Example 2 from Robby

Percentage Accurate: 99.8% → 99.8%

Time: 20.2s

Alternatives: 14

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ \left|\left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(eh \cdot \sin t\right) \cdot \sin t\_1\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* (- eh) (tan t)) ew))))
   (fabs (- (* (* ew (cos t)) (cos t_1)) (* (* eh (sin t)) (sin t_1))))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((-eh * tan(t)) / ew));
	return fabs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))));
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = atan(((-eh * tan(t)) / ew))
    code = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))))
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan(((-eh * Math.tan(t)) / ew));
	return Math.abs((((ew * Math.cos(t)) * Math.cos(t_1)) - ((eh * Math.sin(t)) * Math.sin(t_1))));
}

Python

def code(eh, ew, t):
	t_1 = math.atan(((-eh * math.tan(t)) / ew))
	return math.fabs((((ew * math.cos(t)) * math.cos(t_1)) - ((eh * math.sin(t)) * math.sin(t_1))))

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(Float64(-eh) * tan(t)) / ew))
	return abs(Float64(Float64(Float64(ew * cos(t)) * cos(t_1)) - Float64(Float64(eh * sin(t)) * sin(t_1))))
end

MATLAB

function tmp = code(eh, ew, t)
	t_1 = atan(((-eh * tan(t)) / ew));
	tmp = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))));
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, N[Abs[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ \left|\left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(eh \cdot \sin t\right) \cdot \sin t\_1\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* (- eh) (tan t)) ew))))
   (fabs (- (* (* ew (cos t)) (cos t_1)) (* (* eh (sin t)) (sin t_1))))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((-eh * tan(t)) / ew));
	return fabs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))));
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = atan(((-eh * tan(t)) / ew))
    code = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))))
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan(((-eh * Math.tan(t)) / ew));
	return Math.abs((((ew * Math.cos(t)) * Math.cos(t_1)) - ((eh * Math.sin(t)) * Math.sin(t_1))));
}

Python

def code(eh, ew, t):
	t_1 = math.atan(((-eh * math.tan(t)) / ew))
	return math.fabs((((ew * math.cos(t)) * math.cos(t_1)) - ((eh * math.sin(t)) * math.sin(t_1))))

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(Float64(-eh) * tan(t)) / ew))
	return abs(Float64(Float64(Float64(ew * cos(t)) * cos(t_1)) - Float64(Float64(eh * sin(t)) * sin(t_1))))
end

MATLAB

function tmp = code(eh, ew, t)
	t_1 = atan(((-eh * tan(t)) / ew));
	tmp = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))));
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, N[Abs[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ \left|\left(ew \cdot \cos t\right) \cdot \cos t\_1 - \left(eh \cdot \sin t\right) \cdot \sin t\_1\right| \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (atan (/ (* (- eh) (tan t)) ew))))
   (fabs (- (* (* ew (cos t)) (cos t_1)) (* (* eh (sin t)) (sin t_1))))))

C

double code(double eh, double ew, double t) {
	double t_1 = atan(((-eh * tan(t)) / ew));
	return fabs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))));
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = atan(((-eh * tan(t)) / ew))
    code = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))))
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.atan(((-eh * Math.tan(t)) / ew));
	return Math.abs((((ew * Math.cos(t)) * Math.cos(t_1)) - ((eh * Math.sin(t)) * Math.sin(t_1))));
}

Python

def code(eh, ew, t):
	t_1 = math.atan(((-eh * math.tan(t)) / ew))
	return math.fabs((((ew * math.cos(t)) * math.cos(t_1)) - ((eh * math.sin(t)) * math.sin(t_1))))

Julia

function code(eh, ew, t)
	t_1 = atan(Float64(Float64(Float64(-eh) * tan(t)) / ew))
	return abs(Float64(Float64(Float64(ew * cos(t)) * cos(t_1)) - Float64(Float64(eh * sin(t)) * sin(t_1))))
end

MATLAB

function tmp = code(eh, ew, t)
	t_1 = atan(((-eh * tan(t)) / ew));
	tmp = abs((((ew * cos(t)) * cos(t_1)) - ((eh * sin(t)) * sin(t_1))));
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, N[Abs[N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] * N[Cos[t$95$1], $MachinePrecision]), $MachinePrecision] - N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.8%

   \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 94.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;eh \leq -7.2 \cdot 10^{-141} \lor \neg \left(eh \leq 1.05 \cdot 10^{-106}\right):\\ \;\;\;\;\left|eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{t}{ew}\right) - ew \cdot \left(\cos \tan^{-1} \left(\tan t \cdot \frac{-eh}{ew}\right) \cdot \frac{\cos t}{eh}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= eh -7.2e-141) (not (<= eh 1.05e-106)))
   (fabs
    (*
     eh
     (-
      (* (sin t) (sin (atan (* (- eh) (/ t ew)))))
      (* ew (* (cos (atan (* (tan t) (/ (- eh) ew)))) (/ (cos t) eh))))))
   (fabs (* ew (cos t)))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -7.2e-141) || !(eh <= 1.05e-106)) {
		tmp = fabs((eh * ((sin(t) * sin(atan((-eh * (t / ew))))) - (ew * (cos(atan((tan(t) * (-eh / ew)))) * (cos(t) / eh))))));
	} else {
		tmp = fabs((ew * cos(t)));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((eh <= (-7.2d-141)) .or. (.not. (eh <= 1.05d-106))) then
        tmp = abs((eh * ((sin(t) * sin(atan((-eh * (t / ew))))) - (ew * (cos(atan((tan(t) * (-eh / ew)))) * (cos(t) / eh))))))
    else
        tmp = abs((ew * cos(t)))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((eh <= -7.2e-141) || !(eh <= 1.05e-106)) {
		tmp = Math.abs((eh * ((Math.sin(t) * Math.sin(Math.atan((-eh * (t / ew))))) - (ew * (Math.cos(Math.atan((Math.tan(t) * (-eh / ew)))) * (Math.cos(t) / eh))))));
	} else {
		tmp = Math.abs((ew * Math.cos(t)));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (eh <= -7.2e-141) or not (eh <= 1.05e-106):
		tmp = math.fabs((eh * ((math.sin(t) * math.sin(math.atan((-eh * (t / ew))))) - (ew * (math.cos(math.atan((math.tan(t) * (-eh / ew)))) * (math.cos(t) / eh))))))
	else:
		tmp = math.fabs((ew * math.cos(t)))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((eh <= -7.2e-141) || !(eh <= 1.05e-106))
		tmp = abs(Float64(eh * Float64(Float64(sin(t) * sin(atan(Float64(Float64(-eh) * Float64(t / ew))))) - Float64(ew * Float64(cos(atan(Float64(tan(t) * Float64(Float64(-eh) / ew)))) * Float64(cos(t) / eh))))));
	else
		tmp = abs(Float64(ew * cos(t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((eh <= -7.2e-141) || ~((eh <= 1.05e-106)))
		tmp = abs((eh * ((sin(t) * sin(atan((-eh * (t / ew))))) - (ew * (cos(atan((tan(t) * (-eh / ew)))) * (cos(t) / eh))))));
	else
		tmp = abs((ew * cos(t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[eh, -7.2e-141], N[Not[LessEqual[eh, 1.05e-106]], $MachinePrecision]], N[Abs[N[(eh * N[(N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[N[((-eh) * N[(t / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(ew * N[(N[Cos[N[ArcTan[N[(N[Tan[t], $MachinePrecision] * N[((-eh) / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(N[Cos[t], $MachinePrecision] / eh), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if eh < -7.2000000000000003e-141 or 1.05000000000000002e-106 < eh

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   2. Step-by-step derivation

      1. fabs-sub 99.8%

         \[\leadsto \color{blue}{\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right|} \]

      2. sub-neg 99.8%

         \[\leadsto \left|\color{blue}{\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) + \left(-\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

      3. +-commutative 99.8%

         \[\leadsto \left|\color{blue}{\left(-\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}\right| \]

      4. associate-*l* 99.8%

         \[\leadsto \left|\left(-\color{blue}{ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right) + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      5. distribute-rgt-neg-in 99.8%

         \[\leadsto \left|\color{blue}{ew \cdot \left(-\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)} + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      6. fma-define 99.8%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(ew, -\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right), \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \left(-\cos t\right), eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in eh around inf 98.8%

      \[\leadsto \left|\color{blue}{eh \cdot \left(-1 \cdot \frac{ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{eh} + \sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}\right| \]
   6. Step-by-step derivation

      1. +-commutative 98.8%

         \[\leadsto \left|eh \cdot \color{blue}{\left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right) + -1 \cdot \frac{ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{eh}\right)}\right| \]

      2. mul-1-neg 98.8%

         \[\leadsto \left|eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right) + \color{blue}{\left(-\frac{ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{eh}\right)}\right)\right| \]

      3. unsub-neg 98.8%

         \[\leadsto \left|eh \cdot \color{blue}{\left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right) - \frac{ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}{eh}\right)}\right| \]

   7. Simplified 98.6%

      \[\leadsto \left|\color{blue}{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right) - ew \cdot \left(\cos \tan^{-1} \left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right) \cdot \frac{\cos t}{eh}\right)\right)}\right| \]

   8. Taylor expanded in t around 0 98.1%

      \[\leadsto \left|eh \cdot \left(\sin t \cdot \sin \tan^{-1} \color{blue}{\left(-1 \cdot \frac{eh \cdot t}{ew}\right)} - ew \cdot \left(\cos \tan^{-1} \left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right) \cdot \frac{\cos t}{eh}\right)\right)\right| \]
   9. Step-by-step derivation

      1. mul-1-neg 98.1%

         \[\leadsto \left|eh \cdot \left(\sin t \cdot \sin \tan^{-1} \color{blue}{\left(-\frac{eh \cdot t}{ew}\right)} - ew \cdot \left(\cos \tan^{-1} \left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right) \cdot \frac{\cos t}{eh}\right)\right)\right| \]

      2. associate-/l* 98.1%

         \[\leadsto \left|eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-\color{blue}{eh \cdot \frac{t}{ew}}\right) - ew \cdot \left(\cos \tan^{-1} \left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right) \cdot \frac{\cos t}{eh}\right)\right)\right| \]

      3. distribute-lft-neg-in 98.1%

         \[\leadsto \left|eh \cdot \left(\sin t \cdot \sin \tan^{-1} \color{blue}{\left(\left(-eh\right) \cdot \frac{t}{ew}\right)} - ew \cdot \left(\cos \tan^{-1} \left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right) \cdot \frac{\cos t}{eh}\right)\right)\right| \]

   10. Simplified 98.1%

       \[\leadsto \left|eh \cdot \left(\sin t \cdot \sin \tan^{-1} \color{blue}{\left(\left(-eh\right) \cdot \frac{t}{ew}\right)} - ew \cdot \left(\cos \tan^{-1} \left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right) \cdot \frac{\cos t}{eh}\right)\right)\right| \]

   if -7.2000000000000003e-141 < eh < 1.05000000000000002e-106

   1. Initial program 99.9%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   2. Step-by-step derivation

      1. fabs-sub 99.9%

         \[\leadsto \color{blue}{\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right|} \]

      2. sub-neg 99.9%

         \[\leadsto \left|\color{blue}{\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) + \left(-\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

      3. +-commutative 99.9%

         \[\leadsto \left|\color{blue}{\left(-\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}\right| \]

      4. associate-*l* 99.9%

         \[\leadsto \left|\left(-\color{blue}{ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right) + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      5. distribute-rgt-neg-in 99.9%

         \[\leadsto \left|\color{blue}{ew \cdot \left(-\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)} + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      6. fma-define 99.8%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(ew, -\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right), \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \left(-\cos t\right), eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 22.1%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \color{blue}{\left(\sqrt{-\cos t} \cdot \sqrt{-\cos t}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      2. sqrt-unprod 99.9%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \color{blue}{\sqrt{\left(-\cos t\right) \cdot \left(-\cos t\right)}}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      3. sqr-neg 99.9%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \sqrt{\color{blue}{\cos t \cdot \cos t}}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      4. sqrt-unprod 77.6%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \color{blue}{\left(\sqrt{\cos t} \cdot \sqrt{\cos t}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      5. add-sqr-sqrt 99.9%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \color{blue}{\cos t}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      6. associate-*r/ 99.9%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \color{blue}{\left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)} \cdot \cos t, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      7. add-log-exp 99.8%

         \[\leadsto \left|\mathsf{fma}\left(ew, \color{blue}{\log \left(e^{\cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) \cdot \cos t}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      8. associate-*r/ 99.8%

         \[\leadsto \left|\mathsf{fma}\left(ew, \log \left(e^{\cos \tan^{-1} \color{blue}{\left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)} \cdot \cos t}\right), eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      9. cos-atan 99.8%

         \[\leadsto \left|\mathsf{fma}\left(ew, \log \left(e^{\color{blue}{\frac{1}{\sqrt{1 + \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)}}} \cdot \cos t}\right), eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

   6. Applied egg-rr 99.8%

      \[\leadsto \left|\mathsf{fma}\left(ew, \color{blue}{\log \left(e^{\frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

   7. Taylor expanded in ew around inf 89.5%

      \[\leadsto \left|\color{blue}{ew \cdot \cos t}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;eh \leq -7.2 \cdot 10^{-141} \lor \neg \left(eh \leq 1.05 \cdot 10^{-106}\right):\\ \;\;\;\;\left|eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{t}{ew}\right) - ew \cdot \left(\cos \tan^{-1} \left(\tan t \cdot \frac{-eh}{ew}\right) \cdot \frac{\cos t}{eh}\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\sqrt{\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(t \cdot \frac{eh}{ew}\right)\right|}\right)}^{2}\\ t_2 := \tan^{-1} \left(\tan t \cdot \frac{-eh}{ew}\right)\\ t_3 := \tan t \cdot \frac{eh}{ew}\\ \mathbf{if}\;t \leq -1600000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 6500000:\\ \;\;\;\;\left|\mathsf{fma}\left(ew, \cos t\_2, \left(t \cdot \left(-eh\right)\right) \cdot \sin t\_2\right)\right|\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{+118}:\\ \;\;\;\;\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, t\_3\right)} + \sin t \cdot \left(eh \cdot \sin \tan^{-1} t\_3\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (pow
          (sqrt (fabs (* (* eh (sin t)) (sin (atan (* t (/ eh ew)))))))
          2.0))
        (t_2 (atan (* (tan t) (/ (- eh) ew))))
        (t_3 (* (tan t) (/ eh ew))))
   (if (<= t -1600000000.0)
     t_1
     (if (<= t 6500000.0)
       (fabs (fma ew (cos t_2) (* (* t (- eh)) (sin t_2))))
       (if (<= t 4.4e+118)
         (+
          (/ (* ew (cos t)) (hypot 1.0 t_3))
          (* (sin t) (* eh (sin (atan t_3)))))
         t_1)))))

C

double code(double eh, double ew, double t) {
	double t_1 = pow(sqrt(fabs(((eh * sin(t)) * sin(atan((t * (eh / ew))))))), 2.0);
	double t_2 = atan((tan(t) * (-eh / ew)));
	double t_3 = tan(t) * (eh / ew);
	double tmp;
	if (t <= -1600000000.0) {
		tmp = t_1;
	} else if (t <= 6500000.0) {
		tmp = fabs(fma(ew, cos(t_2), ((t * -eh) * sin(t_2))));
	} else if (t <= 4.4e+118) {
		tmp = ((ew * cos(t)) / hypot(1.0, t_3)) + (sin(t) * (eh * sin(atan(t_3))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(eh, ew, t)
	t_1 = sqrt(abs(Float64(Float64(eh * sin(t)) * sin(atan(Float64(t * Float64(eh / ew))))))) ^ 2.0
	t_2 = atan(Float64(tan(t) * Float64(Float64(-eh) / ew)))
	t_3 = Float64(tan(t) * Float64(eh / ew))
	tmp = 0.0
	if (t <= -1600000000.0)
		tmp = t_1;
	elseif (t <= 6500000.0)
		tmp = abs(fma(ew, cos(t_2), Float64(Float64(t * Float64(-eh)) * sin(t_2))));
	elseif (t <= 4.4e+118)
		tmp = Float64(Float64(Float64(ew * cos(t)) / hypot(1.0, t_3)) + Float64(sin(t) * Float64(eh * sin(atan(t_3)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Power[N[Sqrt[N[Abs[N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(t * N[(eh / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[Tan[t], $MachinePrecision] * N[((-eh) / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Tan[t], $MachinePrecision] * N[(eh / ew), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1600000000.0], t$95$1, If[LessEqual[t, 6500000.0], N[Abs[N[(ew * N[Cos[t$95$2], $MachinePrecision] + N[(N[(t * (-eh)), $MachinePrecision] * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 4.4e+118], N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + t$95$3 ^ 2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[t], $MachinePrecision] * N[(eh * N[Sin[N[ArcTan[t$95$3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.6e9 or 4.39999999999999972e118 < t

   1. Initial program 99.7%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   2. Add Preprocessing

   4. Applied egg-rr 48.5%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(eh \cdot \sin t, \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right), \frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)}\right)}^{2}} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 48.5%

         \[\leadsto {\left(\sqrt{\color{blue}{\sqrt{\mathsf{fma}\left(eh \cdot \sin t, \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right), \frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)} \cdot \sqrt{\mathsf{fma}\left(eh \cdot \sin t, \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right), \frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)}}}\right)}^{2} \]

      2. pow1/2 48.5%

         \[\leadsto {\left(\sqrt{\color{blue}{{\left(\mathsf{fma}\left(eh \cdot \sin t, \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right), \frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)\right)}^{0.5}} \cdot \sqrt{\mathsf{fma}\left(eh \cdot \sin t, \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right), \frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)}}\right)}^{2} \]

      3. pow1/2 48.5%

         \[\leadsto {\left(\sqrt{{\left(\mathsf{fma}\left(eh \cdot \sin t, \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right), \frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)\right)}^{0.5} \cdot \color{blue}{{\left(\mathsf{fma}\left(eh \cdot \sin t, \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right), \frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)\right)}^{0.5}}}\right)}^{2} \]

      4. pow-prod-down 50.3%

         \[\leadsto {\left(\sqrt{\color{blue}{{\left(\mathsf{fma}\left(eh \cdot \sin t, \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right), \frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right) \cdot \mathsf{fma}\left(eh \cdot \sin t, \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right), \frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)\right)}^{0.5}}}\right)}^{2} \]

   6. Applied egg-rr 50.3%

      \[\leadsto {\left(\sqrt{\color{blue}{{\left({\left(\mathsf{fma}\left(eh \cdot \sin t, \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right), \frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)\right)}^{2}\right)}^{0.5}}}\right)}^{2} \]
   7. Step-by-step derivation

      1. unpow1/2 50.3%

         \[\leadsto {\left(\sqrt{\color{blue}{\sqrt{{\left(\mathsf{fma}\left(eh \cdot \sin t, \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right), \frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)\right)}^{2}}}}\right)}^{2} \]

      2. unpow2 50.3%

         \[\leadsto {\left(\sqrt{\sqrt{\color{blue}{\mathsf{fma}\left(eh \cdot \sin t, \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right), \frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right) \cdot \mathsf{fma}\left(eh \cdot \sin t, \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right), \frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)}}}\right)}^{2} \]

      3. rem-sqrt-square 99.2%

         \[\leadsto {\left(\sqrt{\color{blue}{\left|\mathsf{fma}\left(eh \cdot \sin t, \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right), \frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)\right|}}\right)}^{2} \]

   8. Simplified 99.2%

      \[\leadsto {\left(\sqrt{\color{blue}{\left|\mathsf{fma}\left(eh \cdot \sin t, \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right), \frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)\right|}}\right)}^{2} \]

   9. Taylor expanded in t around 0 98.4%

      \[\leadsto {\left(\sqrt{\left|\mathsf{fma}\left(eh \cdot \sin t, \sin \tan^{-1} \color{blue}{\left(\frac{eh \cdot t}{ew}\right)}, \frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)\right|}\right)}^{2} \]

   10. Taylor expanded in eh around inf 60.4%

       \[\leadsto {\left(\sqrt{\left|\color{blue}{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{eh \cdot t}{ew}\right)\right)}\right|}\right)}^{2} \]
   11. Step-by-step derivation

       1. associate-*r/ 60.4%

          \[\leadsto {\left(\sqrt{\left|eh \cdot \left(\sin t \cdot \sin \tan^{-1} \color{blue}{\left(eh \cdot \frac{t}{ew}\right)}\right)\right|}\right)}^{2} \]

       2. associate-*l* 60.4%

          \[\leadsto {\left(\sqrt{\left|\color{blue}{\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(eh \cdot \frac{t}{ew}\right)}\right|}\right)}^{2} \]

       3. *-commutative 60.4%

          \[\leadsto {\left(\sqrt{\left|\color{blue}{\sin \tan^{-1} \left(eh \cdot \frac{t}{ew}\right) \cdot \left(eh \cdot \sin t\right)}\right|}\right)}^{2} \]

       4. associate-*r/ 60.4%

          \[\leadsto {\left(\sqrt{\left|\sin \tan^{-1} \color{blue}{\left(\frac{eh \cdot t}{ew}\right)} \cdot \left(eh \cdot \sin t\right)\right|}\right)}^{2} \]

       5. *-commutative 60.4%

          \[\leadsto {\left(\sqrt{\left|\sin \tan^{-1} \left(\frac{\color{blue}{t \cdot eh}}{ew}\right) \cdot \left(eh \cdot \sin t\right)\right|}\right)}^{2} \]

       6. associate-/l* 60.4%

          \[\leadsto {\left(\sqrt{\left|\sin \tan^{-1} \color{blue}{\left(t \cdot \frac{eh}{ew}\right)} \cdot \left(eh \cdot \sin t\right)\right|}\right)}^{2} \]

   12. Simplified 60.4%

       \[\leadsto {\left(\sqrt{\left|\color{blue}{\sin \tan^{-1} \left(t \cdot \frac{eh}{ew}\right) \cdot \left(eh \cdot \sin t\right)}\right|}\right)}^{2} \]

   if -1.6e9 < t < 6.5e6

   1. Initial program 100.0%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0 97.8%

      \[\leadsto \left|\color{blue}{-1 \cdot \left(eh \cdot \left(t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right) + ew \cdot \cos \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)}\right| \]
   4. Step-by-step derivation

      1. +-commutative 97.8%

         \[\leadsto \left|\color{blue}{ew \cdot \cos \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right) + -1 \cdot \left(eh \cdot \left(t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)}\right| \]

      2. fma-define 97.8%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(ew, \cos \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right), -1 \cdot \left(eh \cdot \left(t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)}\right| \]

      3. associate-*r/ 97.8%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(eh \cdot \tan t\right)}{ew}\right)}, -1 \cdot \left(eh \cdot \left(t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right| \]

      4. neg-mul-1 97.8%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\frac{\color{blue}{-eh \cdot \tan t}}{ew}\right), -1 \cdot \left(eh \cdot \left(t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right| \]

      5. *-commutative 97.8%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\frac{-\color{blue}{\tan t \cdot eh}}{ew}\right), -1 \cdot \left(eh \cdot \left(t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right| \]

      6. distribute-lft-neg-in 97.8%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\frac{\color{blue}{\left(-\tan t\right) \cdot eh}}{ew}\right), -1 \cdot \left(eh \cdot \left(t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right| \]

      7. associate-*r/ 97.8%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \color{blue}{\left(\left(-\tan t\right) \cdot \frac{eh}{ew}\right)}, -1 \cdot \left(eh \cdot \left(t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right| \]

      8. *-commutative 97.8%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \color{blue}{\left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right)}, -1 \cdot \left(eh \cdot \left(t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)\right)\right)\right| \]

      9. mul-1-neg 97.8%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right), \color{blue}{-eh \cdot \left(t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}\right)\right| \]

      10. associate-*r* 97.8%

          \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right), -\color{blue}{\left(eh \cdot t\right) \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)}\right)\right| \]

      11. distribute-lft-neg-in 97.8%

          \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right), \color{blue}{\left(-eh \cdot t\right) \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)}\right)\right| \]

   5. Simplified 97.8%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right), \left(t \cdot \left(-eh\right)\right) \cdot \sin \tan^{-1} \left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right)\right)}\right| \]

   if 6.5e6 < t < 4.39999999999999972e118

   1. Initial program 99.6%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 73.1%

         \[\leadsto \left|\color{blue}{\sqrt{\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)} \cdot \sqrt{\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}}\right| \]

      2. fabs-sqr 73.1%

         \[\leadsto \color{blue}{\sqrt{\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)} \cdot \sqrt{\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}} \]

      3. add-sqr-sqrt 73.6%

         \[\leadsto \color{blue}{\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)} \]

      4. sub-neg 73.6%

         \[\leadsto \color{blue}{\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) + \left(-\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)} \]

   4. Applied egg-rr 73.6%

      \[\leadsto \color{blue}{\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)} + \sin t \cdot \left(eh \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1600000000:\\ \;\;\;\;{\left(\sqrt{\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(t \cdot \frac{eh}{ew}\right)\right|}\right)}^{2}\\ \mathbf{elif}\;t \leq 6500000:\\ \;\;\;\;\left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\tan t \cdot \frac{-eh}{ew}\right), \left(t \cdot \left(-eh\right)\right) \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{-eh}{ew}\right)\right)\right|\\ \mathbf{elif}\;t \leq 4.4 \cdot 10^{+118}:\\ \;\;\;\;\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)} + \sin t \cdot \left(eh \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(t \cdot \frac{eh}{ew}\right)\right|}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := {\left(\sqrt{\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(t \cdot \frac{eh}{ew}\right)\right|}\right)}^{2}\\ t_2 := \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\\ t_3 := \tan t \cdot \frac{eh}{ew}\\ \mathbf{if}\;t \leq -1600000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 6500000:\\ \;\;\;\;\left|eh \cdot \left(t \cdot \sin t\_2\right) - ew \cdot \cos t\_2\right|\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{+117}:\\ \;\;\;\;\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, t\_3\right)} + \sin t \cdot \left(eh \cdot \sin \tan^{-1} t\_3\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1
         (pow
          (sqrt (fabs (* (* eh (sin t)) (sin (atan (* t (/ eh ew)))))))
          2.0))
        (t_2 (atan (/ (* (- eh) (tan t)) ew)))
        (t_3 (* (tan t) (/ eh ew))))
   (if (<= t -1600000000.0)
     t_1
     (if (<= t 6500000.0)
       (fabs (- (* eh (* t (sin t_2))) (* ew (cos t_2))))
       (if (<= t 2.2e+117)
         (+
          (/ (* ew (cos t)) (hypot 1.0 t_3))
          (* (sin t) (* eh (sin (atan t_3)))))
         t_1)))))

C

double code(double eh, double ew, double t) {
	double t_1 = pow(sqrt(fabs(((eh * sin(t)) * sin(atan((t * (eh / ew))))))), 2.0);
	double t_2 = atan(((-eh * tan(t)) / ew));
	double t_3 = tan(t) * (eh / ew);
	double tmp;
	if (t <= -1600000000.0) {
		tmp = t_1;
	} else if (t <= 6500000.0) {
		tmp = fabs(((eh * (t * sin(t_2))) - (ew * cos(t_2))));
	} else if (t <= 2.2e+117) {
		tmp = ((ew * cos(t)) / hypot(1.0, t_3)) + (sin(t) * (eh * sin(atan(t_3))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.pow(Math.sqrt(Math.abs(((eh * Math.sin(t)) * Math.sin(Math.atan((t * (eh / ew))))))), 2.0);
	double t_2 = Math.atan(((-eh * Math.tan(t)) / ew));
	double t_3 = Math.tan(t) * (eh / ew);
	double tmp;
	if (t <= -1600000000.0) {
		tmp = t_1;
	} else if (t <= 6500000.0) {
		tmp = Math.abs(((eh * (t * Math.sin(t_2))) - (ew * Math.cos(t_2))));
	} else if (t <= 2.2e+117) {
		tmp = ((ew * Math.cos(t)) / Math.hypot(1.0, t_3)) + (Math.sin(t) * (eh * Math.sin(Math.atan(t_3))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.pow(math.sqrt(math.fabs(((eh * math.sin(t)) * math.sin(math.atan((t * (eh / ew))))))), 2.0)
	t_2 = math.atan(((-eh * math.tan(t)) / ew))
	t_3 = math.tan(t) * (eh / ew)
	tmp = 0
	if t <= -1600000000.0:
		tmp = t_1
	elif t <= 6500000.0:
		tmp = math.fabs(((eh * (t * math.sin(t_2))) - (ew * math.cos(t_2))))
	elif t <= 2.2e+117:
		tmp = ((ew * math.cos(t)) / math.hypot(1.0, t_3)) + (math.sin(t) * (eh * math.sin(math.atan(t_3))))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = sqrt(abs(Float64(Float64(eh * sin(t)) * sin(atan(Float64(t * Float64(eh / ew))))))) ^ 2.0
	t_2 = atan(Float64(Float64(Float64(-eh) * tan(t)) / ew))
	t_3 = Float64(tan(t) * Float64(eh / ew))
	tmp = 0.0
	if (t <= -1600000000.0)
		tmp = t_1;
	elseif (t <= 6500000.0)
		tmp = abs(Float64(Float64(eh * Float64(t * sin(t_2))) - Float64(ew * cos(t_2))));
	elseif (t <= 2.2e+117)
		tmp = Float64(Float64(Float64(ew * cos(t)) / hypot(1.0, t_3)) + Float64(sin(t) * Float64(eh * sin(atan(t_3)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = sqrt(abs(((eh * sin(t)) * sin(atan((t * (eh / ew))))))) ^ 2.0;
	t_2 = atan(((-eh * tan(t)) / ew));
	t_3 = tan(t) * (eh / ew);
	tmp = 0.0;
	if (t <= -1600000000.0)
		tmp = t_1;
	elseif (t <= 6500000.0)
		tmp = abs(((eh * (t * sin(t_2))) - (ew * cos(t_2))));
	elseif (t <= 2.2e+117)
		tmp = ((ew * cos(t)) / hypot(1.0, t_3)) + (sin(t) * (eh * sin(atan(t_3))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Power[N[Sqrt[N[Abs[N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(t * N[(eh / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[ArcTan[N[(N[((-eh) * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Tan[t], $MachinePrecision] * N[(eh / ew), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1600000000.0], t$95$1, If[LessEqual[t, 6500000.0], N[Abs[N[(N[(eh * N[(t * N[Sin[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(ew * N[Cos[t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[t, 2.2e+117], N[(N[(N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision] / N[Sqrt[1.0 ^ 2 + t$95$3 ^ 2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[t], $MachinePrecision] * N[(eh * N[Sin[N[ArcTan[t$95$3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.6e9 or 2.20000000000000014e117 < t

   1. Initial program 99.7%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   2. Add Preprocessing

   4. Applied egg-rr 48.5%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(eh \cdot \sin t, \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right), \frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)}\right)}^{2}} \]
   5. Step-by-step derivation

      1. add-sqr-sqrt 48.5%

         \[\leadsto {\left(\sqrt{\color{blue}{\sqrt{\mathsf{fma}\left(eh \cdot \sin t, \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right), \frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)} \cdot \sqrt{\mathsf{fma}\left(eh \cdot \sin t, \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right), \frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)}}}\right)}^{2} \]

      2. pow1/2 48.5%

         \[\leadsto {\left(\sqrt{\color{blue}{{\left(\mathsf{fma}\left(eh \cdot \sin t, \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right), \frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)\right)}^{0.5}} \cdot \sqrt{\mathsf{fma}\left(eh \cdot \sin t, \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right), \frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)}}\right)}^{2} \]

      3. pow1/2 48.5%

         \[\leadsto {\left(\sqrt{{\left(\mathsf{fma}\left(eh \cdot \sin t, \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right), \frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)\right)}^{0.5} \cdot \color{blue}{{\left(\mathsf{fma}\left(eh \cdot \sin t, \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right), \frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)\right)}^{0.5}}}\right)}^{2} \]

      4. pow-prod-down 50.3%

         \[\leadsto {\left(\sqrt{\color{blue}{{\left(\mathsf{fma}\left(eh \cdot \sin t, \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right), \frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right) \cdot \mathsf{fma}\left(eh \cdot \sin t, \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right), \frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)\right)}^{0.5}}}\right)}^{2} \]

   6. Applied egg-rr 50.3%

      \[\leadsto {\left(\sqrt{\color{blue}{{\left({\left(\mathsf{fma}\left(eh \cdot \sin t, \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right), \frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)\right)}^{2}\right)}^{0.5}}}\right)}^{2} \]
   7. Step-by-step derivation

      1. unpow1/2 50.3%

         \[\leadsto {\left(\sqrt{\color{blue}{\sqrt{{\left(\mathsf{fma}\left(eh \cdot \sin t, \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right), \frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)\right)}^{2}}}}\right)}^{2} \]

      2. unpow2 50.3%

         \[\leadsto {\left(\sqrt{\sqrt{\color{blue}{\mathsf{fma}\left(eh \cdot \sin t, \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right), \frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right) \cdot \mathsf{fma}\left(eh \cdot \sin t, \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right), \frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)}}}\right)}^{2} \]

      3. rem-sqrt-square 99.2%

         \[\leadsto {\left(\sqrt{\color{blue}{\left|\mathsf{fma}\left(eh \cdot \sin t, \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right), \frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)\right|}}\right)}^{2} \]

   8. Simplified 99.2%

      \[\leadsto {\left(\sqrt{\color{blue}{\left|\mathsf{fma}\left(eh \cdot \sin t, \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right), \frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)\right|}}\right)}^{2} \]

   9. Taylor expanded in t around 0 98.4%

      \[\leadsto {\left(\sqrt{\left|\mathsf{fma}\left(eh \cdot \sin t, \sin \tan^{-1} \color{blue}{\left(\frac{eh \cdot t}{ew}\right)}, \frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)\right|}\right)}^{2} \]

   10. Taylor expanded in eh around inf 60.4%

       \[\leadsto {\left(\sqrt{\left|\color{blue}{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{eh \cdot t}{ew}\right)\right)}\right|}\right)}^{2} \]
   11. Step-by-step derivation

       1. associate-*r/ 60.4%

          \[\leadsto {\left(\sqrt{\left|eh \cdot \left(\sin t \cdot \sin \tan^{-1} \color{blue}{\left(eh \cdot \frac{t}{ew}\right)}\right)\right|}\right)}^{2} \]

       2. associate-*l* 60.4%

          \[\leadsto {\left(\sqrt{\left|\color{blue}{\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(eh \cdot \frac{t}{ew}\right)}\right|}\right)}^{2} \]

       3. *-commutative 60.4%

          \[\leadsto {\left(\sqrt{\left|\color{blue}{\sin \tan^{-1} \left(eh \cdot \frac{t}{ew}\right) \cdot \left(eh \cdot \sin t\right)}\right|}\right)}^{2} \]

       4. associate-*r/ 60.4%

          \[\leadsto {\left(\sqrt{\left|\sin \tan^{-1} \color{blue}{\left(\frac{eh \cdot t}{ew}\right)} \cdot \left(eh \cdot \sin t\right)\right|}\right)}^{2} \]

       5. *-commutative 60.4%

          \[\leadsto {\left(\sqrt{\left|\sin \tan^{-1} \left(\frac{\color{blue}{t \cdot eh}}{ew}\right) \cdot \left(eh \cdot \sin t\right)\right|}\right)}^{2} \]

       6. associate-/l* 60.4%

          \[\leadsto {\left(\sqrt{\left|\sin \tan^{-1} \color{blue}{\left(t \cdot \frac{eh}{ew}\right)} \cdot \left(eh \cdot \sin t\right)\right|}\right)}^{2} \]

   12. Simplified 60.4%

       \[\leadsto {\left(\sqrt{\left|\color{blue}{\sin \tan^{-1} \left(t \cdot \frac{eh}{ew}\right) \cdot \left(eh \cdot \sin t\right)}\right|}\right)}^{2} \]

   if -1.6e9 < t < 6.5e6

   1. Initial program 100.0%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   2. Step-by-step derivation

      1. fabs-sub 100.0%

         \[\leadsto \color{blue}{\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right|} \]

      2. sub-neg 100.0%

         \[\leadsto \left|\color{blue}{\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) + \left(-\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

      3. +-commutative 100.0%

         \[\leadsto \left|\color{blue}{\left(-\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}\right| \]

      4. associate-*l* 100.0%

         \[\leadsto \left|\left(-\color{blue}{ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right) + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      5. distribute-rgt-neg-in 100.0%

         \[\leadsto \left|\color{blue}{ew \cdot \left(-\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)} + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      6. fma-define 100.0%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(ew, -\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right), \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \left(-\cos t\right), eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 97.8%

      \[\leadsto \left|\color{blue}{-1 \cdot \left(ew \cdot \cos \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right) + eh \cdot \left(t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}\right| \]

   if 6.5e6 < t < 2.20000000000000014e117

   1. Initial program 99.6%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 73.1%

         \[\leadsto \left|\color{blue}{\sqrt{\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)} \cdot \sqrt{\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}}\right| \]

      2. fabs-sqr 73.1%

         \[\leadsto \color{blue}{\sqrt{\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)} \cdot \sqrt{\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}} \]

      3. add-sqr-sqrt 73.6%

         \[\leadsto \color{blue}{\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)} \]

      4. sub-neg 73.6%

         \[\leadsto \color{blue}{\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) + \left(-\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)} \]

   4. Applied egg-rr 73.6%

      \[\leadsto \color{blue}{\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)} + \sin t \cdot \left(eh \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1600000000:\\ \;\;\;\;{\left(\sqrt{\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(t \cdot \frac{eh}{ew}\right)\right|}\right)}^{2}\\ \mathbf{elif}\;t \leq 6500000:\\ \;\;\;\;\left|eh \cdot \left(t \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) - ew \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right|\\ \mathbf{elif}\;t \leq 2.2 \cdot 10^{+117}:\\ \;\;\;\;\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)} + \sin t \cdot \left(eh \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt{\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(t \cdot \frac{eh}{ew}\right)\right|}\right)}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \tan t \cdot \frac{eh}{ew}\\ t_2 := ew \cdot \cos t\\ t_3 := \left|t\_2\right|\\ \mathbf{if}\;ew \leq -3.2 \cdot 10^{-104}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;ew \leq 8.5 \cdot 10^{-213}:\\ \;\;\;\;\left|\sin t \cdot \left(eh \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{-eh}{ew}\right)\right)\right|\\ \mathbf{elif}\;ew \leq 1.4 \cdot 10^{+114}:\\ \;\;\;\;\frac{t\_2}{\mathsf{hypot}\left(1, t\_1\right)} + \sin t \cdot \left(eh \cdot \sin \tan^{-1} t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* (tan t) (/ eh ew))) (t_2 (* ew (cos t))) (t_3 (fabs t_2)))
   (if (<= ew -3.2e-104)
     t_3
     (if (<= ew 8.5e-213)
       (fabs (* (sin t) (* eh (sin (atan (* (tan t) (/ (- eh) ew)))))))
       (if (<= ew 1.4e+114)
         (+ (/ t_2 (hypot 1.0 t_1)) (* (sin t) (* eh (sin (atan t_1)))))
         t_3)))))

C

double code(double eh, double ew, double t) {
	double t_1 = tan(t) * (eh / ew);
	double t_2 = ew * cos(t);
	double t_3 = fabs(t_2);
	double tmp;
	if (ew <= -3.2e-104) {
		tmp = t_3;
	} else if (ew <= 8.5e-213) {
		tmp = fabs((sin(t) * (eh * sin(atan((tan(t) * (-eh / ew)))))));
	} else if (ew <= 1.4e+114) {
		tmp = (t_2 / hypot(1.0, t_1)) + (sin(t) * (eh * sin(atan(t_1))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.tan(t) * (eh / ew);
	double t_2 = ew * Math.cos(t);
	double t_3 = Math.abs(t_2);
	double tmp;
	if (ew <= -3.2e-104) {
		tmp = t_3;
	} else if (ew <= 8.5e-213) {
		tmp = Math.abs((Math.sin(t) * (eh * Math.sin(Math.atan((Math.tan(t) * (-eh / ew)))))));
	} else if (ew <= 1.4e+114) {
		tmp = (t_2 / Math.hypot(1.0, t_1)) + (Math.sin(t) * (eh * Math.sin(Math.atan(t_1))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.tan(t) * (eh / ew)
	t_2 = ew * math.cos(t)
	t_3 = math.fabs(t_2)
	tmp = 0
	if ew <= -3.2e-104:
		tmp = t_3
	elif ew <= 8.5e-213:
		tmp = math.fabs((math.sin(t) * (eh * math.sin(math.atan((math.tan(t) * (-eh / ew)))))))
	elif ew <= 1.4e+114:
		tmp = (t_2 / math.hypot(1.0, t_1)) + (math.sin(t) * (eh * math.sin(math.atan(t_1))))
	else:
		tmp = t_3
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(tan(t) * Float64(eh / ew))
	t_2 = Float64(ew * cos(t))
	t_3 = abs(t_2)
	tmp = 0.0
	if (ew <= -3.2e-104)
		tmp = t_3;
	elseif (ew <= 8.5e-213)
		tmp = abs(Float64(sin(t) * Float64(eh * sin(atan(Float64(tan(t) * Float64(Float64(-eh) / ew)))))));
	elseif (ew <= 1.4e+114)
		tmp = Float64(Float64(t_2 / hypot(1.0, t_1)) + Float64(sin(t) * Float64(eh * sin(atan(t_1)))));
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = tan(t) * (eh / ew);
	t_2 = ew * cos(t);
	t_3 = abs(t_2);
	tmp = 0.0;
	if (ew <= -3.2e-104)
		tmp = t_3;
	elseif (ew <= 8.5e-213)
		tmp = abs((sin(t) * (eh * sin(atan((tan(t) * (-eh / ew)))))));
	elseif (ew <= 1.4e+114)
		tmp = (t_2 / hypot(1.0, t_1)) + (sin(t) * (eh * sin(atan(t_1))));
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(N[Tan[t], $MachinePrecision] * N[(eh / ew), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Abs[t$95$2], $MachinePrecision]}, If[LessEqual[ew, -3.2e-104], t$95$3, If[LessEqual[ew, 8.5e-213], N[Abs[N[(N[Sin[t], $MachinePrecision] * N[(eh * N[Sin[N[ArcTan[N[(N[Tan[t], $MachinePrecision] * N[((-eh) / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 1.4e+114], N[(N[(t$95$2 / N[Sqrt[1.0 ^ 2 + t$95$1 ^ 2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[t], $MachinePrecision] * N[(eh * N[Sin[N[ArcTan[t$95$1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -3.19999999999999989e-104 or 1.4e114 < ew

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   2. Step-by-step derivation

      1. fabs-sub 99.8%

         \[\leadsto \color{blue}{\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right|} \]

      2. sub-neg 99.8%

         \[\leadsto \left|\color{blue}{\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) + \left(-\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

      3. +-commutative 99.8%

         \[\leadsto \left|\color{blue}{\left(-\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}\right| \]

      4. associate-*l* 99.8%

         \[\leadsto \left|\left(-\color{blue}{ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right) + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      5. distribute-rgt-neg-in 99.8%

         \[\leadsto \left|\color{blue}{ew \cdot \left(-\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)} + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      6. fma-define 99.8%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(ew, -\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right), \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \left(-\cos t\right), eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 21.7%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \color{blue}{\left(\sqrt{-\cos t} \cdot \sqrt{-\cos t}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      2. sqrt-unprod 99.2%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \color{blue}{\sqrt{\left(-\cos t\right) \cdot \left(-\cos t\right)}}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      3. sqr-neg 99.2%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \sqrt{\color{blue}{\cos t \cdot \cos t}}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      4. sqrt-unprod 77.3%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \color{blue}{\left(\sqrt{\cos t} \cdot \sqrt{\cos t}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      5. add-sqr-sqrt 99.1%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \color{blue}{\cos t}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      6. associate-*r/ 99.1%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \color{blue}{\left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)} \cdot \cos t, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      7. add-log-exp 98.9%

         \[\leadsto \left|\mathsf{fma}\left(ew, \color{blue}{\log \left(e^{\cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) \cdot \cos t}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      8. associate-*r/ 98.9%

         \[\leadsto \left|\mathsf{fma}\left(ew, \log \left(e^{\cos \tan^{-1} \color{blue}{\left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)} \cdot \cos t}\right), eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      9. cos-atan 98.9%

         \[\leadsto \left|\mathsf{fma}\left(ew, \log \left(e^{\color{blue}{\frac{1}{\sqrt{1 + \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)}}} \cdot \cos t}\right), eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

   6. Applied egg-rr 98.9%

      \[\leadsto \left|\mathsf{fma}\left(ew, \color{blue}{\log \left(e^{\frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

   7. Taylor expanded in ew around inf 82.9%

      \[\leadsto \left|\color{blue}{ew \cdot \cos t}\right| \]

   if -3.19999999999999989e-104 < ew < 8.49999999999999994e-213

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   2. Step-by-step derivation

      1. fabs-sub 99.8%

         \[\leadsto \color{blue}{\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right|} \]

      2. sub-neg 99.8%

         \[\leadsto \left|\color{blue}{\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) + \left(-\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

      3. +-commutative 99.8%

         \[\leadsto \left|\color{blue}{\left(-\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}\right| \]

      4. associate-*l* 99.8%

         \[\leadsto \left|\left(-\color{blue}{ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right) + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      5. distribute-rgt-neg-in 99.8%

         \[\leadsto \left|\color{blue}{ew \cdot \left(-\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)} + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      6. fma-define 99.8%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(ew, -\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right), \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \left(-\cos t\right), eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in ew around 0 84.0%

      \[\leadsto \left|\color{blue}{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}\right| \]
   6. Step-by-step derivation

      1. *-commutative 84.0%

         \[\leadsto \left|\color{blue}{\left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right) \cdot eh}\right| \]

      2. associate-*l* 84.0%

         \[\leadsto \left|\color{blue}{\sin t \cdot \left(\sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right) \cdot eh\right)}\right| \]

      3. *-commutative 84.0%

         \[\leadsto \left|\sin t \cdot \color{blue}{\left(eh \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}\right| \]

      4. associate-*r/ 84.0%

         \[\leadsto \left|\sin t \cdot \left(eh \cdot \sin \tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(eh \cdot \tan t\right)}{ew}\right)}\right)\right| \]

      5. neg-mul-1 84.0%

         \[\leadsto \left|\sin t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{\color{blue}{-eh \cdot \tan t}}{ew}\right)\right)\right| \]

      6. *-commutative 84.0%

         \[\leadsto \left|\sin t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{-\color{blue}{\tan t \cdot eh}}{ew}\right)\right)\right| \]

      7. distribute-lft-neg-in 84.0%

         \[\leadsto \left|\sin t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\left(-\tan t\right) \cdot eh}}{ew}\right)\right)\right| \]

      8. associate-*r/ 84.0%

         \[\leadsto \left|\sin t \cdot \left(eh \cdot \sin \tan^{-1} \color{blue}{\left(\left(-\tan t\right) \cdot \frac{eh}{ew}\right)}\right)\right| \]

      9. *-commutative 84.0%

         \[\leadsto \left|\sin t \cdot \left(eh \cdot \sin \tan^{-1} \color{blue}{\left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right)}\right)\right| \]

   7. Simplified 84.0%

      \[\leadsto \left|\color{blue}{\sin t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right)\right)}\right| \]

   if 8.49999999999999994e-213 < ew < 1.4e114

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. add-sqr-sqrt 74.4%

         \[\leadsto \left|\color{blue}{\sqrt{\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)} \cdot \sqrt{\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}}\right| \]

      2. fabs-sqr 74.4%

         \[\leadsto \color{blue}{\sqrt{\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)} \cdot \sqrt{\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}} \]

      3. add-sqr-sqrt 75.2%

         \[\leadsto \color{blue}{\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)} \]

      4. sub-neg 75.2%

         \[\leadsto \color{blue}{\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) + \left(-\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)} \]

   4. Applied egg-rr 75.2%

      \[\leadsto \color{blue}{\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)} + \sin t \cdot \left(eh \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -3.2 \cdot 10^{-104}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{elif}\;ew \leq 8.5 \cdot 10^{-213}:\\ \;\;\;\;\left|\sin t \cdot \left(eh \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{-eh}{ew}\right)\right)\right|\\ \mathbf{elif}\;ew \leq 1.4 \cdot 10^{+114}:\\ \;\;\;\;\frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)} + \sin t \cdot \left(eh \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := ew \cdot \cos t\\ t_2 := \left|t\_1\right|\\ \mathbf{if}\;ew \leq -1.45 \cdot 10^{-107}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;ew \leq 8.2 \cdot 10^{-213}:\\ \;\;\;\;\left|\sin t \cdot \left(eh \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{-eh}{ew}\right)\right)\right|\\ \mathbf{elif}\;ew \leq 3.7 \cdot 10^{+114}:\\ \;\;\;\;t\_1 + eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{ew}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (* ew (cos t))) (t_2 (fabs t_1)))
   (if (<= ew -1.45e-107)
     t_2
     (if (<= ew 8.2e-213)
       (fabs (* (sin t) (* eh (sin (atan (* (tan t) (/ (- eh) ew)))))))
       (if (<= ew 3.7e+114)
         (+ t_1 (* eh (* (sin t) (sin (atan (/ (* eh (tan t)) ew))))))
         t_2)))))

C

double code(double eh, double ew, double t) {
	double t_1 = ew * cos(t);
	double t_2 = fabs(t_1);
	double tmp;
	if (ew <= -1.45e-107) {
		tmp = t_2;
	} else if (ew <= 8.2e-213) {
		tmp = fabs((sin(t) * (eh * sin(atan((tan(t) * (-eh / ew)))))));
	} else if (ew <= 3.7e+114) {
		tmp = t_1 + (eh * (sin(t) * sin(atan(((eh * tan(t)) / ew)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = ew * cos(t)
    t_2 = abs(t_1)
    if (ew <= (-1.45d-107)) then
        tmp = t_2
    else if (ew <= 8.2d-213) then
        tmp = abs((sin(t) * (eh * sin(atan((tan(t) * (-eh / ew)))))))
    else if (ew <= 3.7d+114) then
        tmp = t_1 + (eh * (sin(t) * sin(atan(((eh * tan(t)) / ew)))))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = ew * Math.cos(t);
	double t_2 = Math.abs(t_1);
	double tmp;
	if (ew <= -1.45e-107) {
		tmp = t_2;
	} else if (ew <= 8.2e-213) {
		tmp = Math.abs((Math.sin(t) * (eh * Math.sin(Math.atan((Math.tan(t) * (-eh / ew)))))));
	} else if (ew <= 3.7e+114) {
		tmp = t_1 + (eh * (Math.sin(t) * Math.sin(Math.atan(((eh * Math.tan(t)) / ew)))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = ew * math.cos(t)
	t_2 = math.fabs(t_1)
	tmp = 0
	if ew <= -1.45e-107:
		tmp = t_2
	elif ew <= 8.2e-213:
		tmp = math.fabs((math.sin(t) * (eh * math.sin(math.atan((math.tan(t) * (-eh / ew)))))))
	elif ew <= 3.7e+114:
		tmp = t_1 + (eh * (math.sin(t) * math.sin(math.atan(((eh * math.tan(t)) / ew)))))
	else:
		tmp = t_2
	return tmp

Julia

function code(eh, ew, t)
	t_1 = Float64(ew * cos(t))
	t_2 = abs(t_1)
	tmp = 0.0
	if (ew <= -1.45e-107)
		tmp = t_2;
	elseif (ew <= 8.2e-213)
		tmp = abs(Float64(sin(t) * Float64(eh * sin(atan(Float64(tan(t) * Float64(Float64(-eh) / ew)))))));
	elseif (ew <= 3.7e+114)
		tmp = Float64(t_1 + Float64(eh * Float64(sin(t) * sin(atan(Float64(Float64(eh * tan(t)) / ew))))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = ew * cos(t);
	t_2 = abs(t_1);
	tmp = 0.0;
	if (ew <= -1.45e-107)
		tmp = t_2;
	elseif (ew <= 8.2e-213)
		tmp = abs((sin(t) * (eh * sin(atan((tan(t) * (-eh / ew)))))));
	elseif (ew <= 3.7e+114)
		tmp = t_1 + (eh * (sin(t) * sin(atan(((eh * tan(t)) / ew)))));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Abs[t$95$1], $MachinePrecision]}, If[LessEqual[ew, -1.45e-107], t$95$2, If[LessEqual[ew, 8.2e-213], N[Abs[N[(N[Sin[t], $MachinePrecision] * N[(eh * N[Sin[N[ArcTan[N[(N[Tan[t], $MachinePrecision] * N[((-eh) / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 3.7e+114], N[(t$95$1 + N[(eh * N[(N[Sin[t], $MachinePrecision] * N[Sin[N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -1.4499999999999999e-107 or 3.7000000000000001e114 < ew

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   2. Step-by-step derivation

      1. fabs-sub 99.8%

         \[\leadsto \color{blue}{\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right|} \]

      2. sub-neg 99.8%

         \[\leadsto \left|\color{blue}{\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) + \left(-\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

      3. +-commutative 99.8%

         \[\leadsto \left|\color{blue}{\left(-\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}\right| \]

      4. associate-*l* 99.8%

         \[\leadsto \left|\left(-\color{blue}{ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right) + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      5. distribute-rgt-neg-in 99.8%

         \[\leadsto \left|\color{blue}{ew \cdot \left(-\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)} + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      6. fma-define 99.8%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(ew, -\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right), \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \left(-\cos t\right), eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 21.7%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \color{blue}{\left(\sqrt{-\cos t} \cdot \sqrt{-\cos t}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      2. sqrt-unprod 99.2%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \color{blue}{\sqrt{\left(-\cos t\right) \cdot \left(-\cos t\right)}}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      3. sqr-neg 99.2%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \sqrt{\color{blue}{\cos t \cdot \cos t}}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      4. sqrt-unprod 77.3%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \color{blue}{\left(\sqrt{\cos t} \cdot \sqrt{\cos t}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      5. add-sqr-sqrt 99.1%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \color{blue}{\cos t}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      6. associate-*r/ 99.1%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \color{blue}{\left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)} \cdot \cos t, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      7. add-log-exp 98.9%

         \[\leadsto \left|\mathsf{fma}\left(ew, \color{blue}{\log \left(e^{\cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) \cdot \cos t}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      8. associate-*r/ 98.9%

         \[\leadsto \left|\mathsf{fma}\left(ew, \log \left(e^{\cos \tan^{-1} \color{blue}{\left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)} \cdot \cos t}\right), eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      9. cos-atan 98.9%

         \[\leadsto \left|\mathsf{fma}\left(ew, \log \left(e^{\color{blue}{\frac{1}{\sqrt{1 + \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)}}} \cdot \cos t}\right), eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

   6. Applied egg-rr 98.9%

      \[\leadsto \left|\mathsf{fma}\left(ew, \color{blue}{\log \left(e^{\frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

   7. Taylor expanded in ew around inf 82.9%

      \[\leadsto \left|\color{blue}{ew \cdot \cos t}\right| \]

   if -1.4499999999999999e-107 < ew < 8.19999999999999949e-213

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   2. Step-by-step derivation

      1. fabs-sub 99.8%

         \[\leadsto \color{blue}{\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right|} \]

      2. sub-neg 99.8%

         \[\leadsto \left|\color{blue}{\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) + \left(-\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

      3. +-commutative 99.8%

         \[\leadsto \left|\color{blue}{\left(-\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}\right| \]

      4. associate-*l* 99.8%

         \[\leadsto \left|\left(-\color{blue}{ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right) + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      5. distribute-rgt-neg-in 99.8%

         \[\leadsto \left|\color{blue}{ew \cdot \left(-\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)} + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      6. fma-define 99.8%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(ew, -\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right), \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \left(-\cos t\right), eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in ew around 0 84.0%

      \[\leadsto \left|\color{blue}{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}\right| \]
   6. Step-by-step derivation

      1. *-commutative 84.0%

         \[\leadsto \left|\color{blue}{\left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right) \cdot eh}\right| \]

      2. associate-*l* 84.0%

         \[\leadsto \left|\color{blue}{\sin t \cdot \left(\sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right) \cdot eh\right)}\right| \]

      3. *-commutative 84.0%

         \[\leadsto \left|\sin t \cdot \color{blue}{\left(eh \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}\right| \]

      4. associate-*r/ 84.0%

         \[\leadsto \left|\sin t \cdot \left(eh \cdot \sin \tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(eh \cdot \tan t\right)}{ew}\right)}\right)\right| \]

      5. neg-mul-1 84.0%

         \[\leadsto \left|\sin t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{\color{blue}{-eh \cdot \tan t}}{ew}\right)\right)\right| \]

      6. *-commutative 84.0%

         \[\leadsto \left|\sin t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{-\color{blue}{\tan t \cdot eh}}{ew}\right)\right)\right| \]

      7. distribute-lft-neg-in 84.0%

         \[\leadsto \left|\sin t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\left(-\tan t\right) \cdot eh}}{ew}\right)\right)\right| \]

      8. associate-*r/ 84.0%

         \[\leadsto \left|\sin t \cdot \left(eh \cdot \sin \tan^{-1} \color{blue}{\left(\left(-\tan t\right) \cdot \frac{eh}{ew}\right)}\right)\right| \]

      9. *-commutative 84.0%

         \[\leadsto \left|\sin t \cdot \left(eh \cdot \sin \tan^{-1} \color{blue}{\left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right)}\right)\right| \]

   7. Simplified 84.0%

      \[\leadsto \left|\color{blue}{\sin t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right)\right)}\right| \]

   if 8.19999999999999949e-213 < ew < 3.7000000000000001e114

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   2. Add Preprocessing

   4. Applied egg-rr 74.4%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(eh \cdot \sin t, \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right), \frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)}\right)}^{2}} \]

   5. Taylor expanded in eh around 0 75.2%

      \[\leadsto \color{blue}{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{ew}\right)\right) + ew \cdot \cos t} \]
3. Recombined 3 regimes into one program.

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -1.45 \cdot 10^{-107}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{elif}\;ew \leq 8.2 \cdot 10^{-213}:\\ \;\;\;\;\left|\sin t \cdot \left(eh \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{-eh}{ew}\right)\right)\right|\\ \mathbf{elif}\;ew \leq 3.7 \cdot 10^{+114}:\\ \;\;\;\;ew \cdot \cos t + eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{ew}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 75.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -5.2 \cdot 10^{-104} \lor \neg \left(ew \leq 6.4 \cdot 10^{-49}\right):\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin t \cdot \left(eh \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{-eh}{ew}\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -5.2e-104) (not (<= ew 6.4e-49)))
   (fabs (* ew (cos t)))
   (fabs (* (sin t) (* eh (sin (atan (* (tan t) (/ (- eh) ew)))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -5.2e-104) || !(ew <= 6.4e-49)) {
		tmp = fabs((ew * cos(t)));
	} else {
		tmp = fabs((sin(t) * (eh * sin(atan((tan(t) * (-eh / ew)))))));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((ew <= (-5.2d-104)) .or. (.not. (ew <= 6.4d-49))) then
        tmp = abs((ew * cos(t)))
    else
        tmp = abs((sin(t) * (eh * sin(atan((tan(t) * (-eh / ew)))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -5.2e-104) || !(ew <= 6.4e-49)) {
		tmp = Math.abs((ew * Math.cos(t)));
	} else {
		tmp = Math.abs((Math.sin(t) * (eh * Math.sin(Math.atan((Math.tan(t) * (-eh / ew)))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -5.2e-104) or not (ew <= 6.4e-49):
		tmp = math.fabs((ew * math.cos(t)))
	else:
		tmp = math.fabs((math.sin(t) * (eh * math.sin(math.atan((math.tan(t) * (-eh / ew)))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -5.2e-104) || !(ew <= 6.4e-49))
		tmp = abs(Float64(ew * cos(t)));
	else
		tmp = abs(Float64(sin(t) * Float64(eh * sin(atan(Float64(tan(t) * Float64(Float64(-eh) / ew)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -5.2e-104) || ~((ew <= 6.4e-49)))
		tmp = abs((ew * cos(t)));
	else
		tmp = abs((sin(t) * (eh * sin(atan((tan(t) * (-eh / ew)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -5.2e-104], N[Not[LessEqual[ew, 6.4e-49]], $MachinePrecision]], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Sin[t], $MachinePrecision] * N[(eh * N[Sin[N[ArcTan[N[(N[Tan[t], $MachinePrecision] * N[((-eh) / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -5.20000000000000005e-104 or 6.40000000000000005e-49 < ew

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   2. Step-by-step derivation

      1. fabs-sub 99.8%

         \[\leadsto \color{blue}{\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right|} \]

      2. sub-neg 99.8%

         \[\leadsto \left|\color{blue}{\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) + \left(-\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

      3. +-commutative 99.8%

         \[\leadsto \left|\color{blue}{\left(-\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}\right| \]

      4. associate-*l* 99.8%

         \[\leadsto \left|\left(-\color{blue}{ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right) + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      5. distribute-rgt-neg-in 99.8%

         \[\leadsto \left|\color{blue}{ew \cdot \left(-\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)} + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      6. fma-define 99.8%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(ew, -\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right), \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \left(-\cos t\right), eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 22.3%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \color{blue}{\left(\sqrt{-\cos t} \cdot \sqrt{-\cos t}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      2. sqrt-unprod 99.3%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \color{blue}{\sqrt{\left(-\cos t\right) \cdot \left(-\cos t\right)}}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      3. sqr-neg 99.3%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \sqrt{\color{blue}{\cos t \cdot \cos t}}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      4. sqrt-unprod 76.9%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \color{blue}{\left(\sqrt{\cos t} \cdot \sqrt{\cos t}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      5. add-sqr-sqrt 99.3%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \color{blue}{\cos t}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      6. associate-*r/ 99.3%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \color{blue}{\left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)} \cdot \cos t, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      7. add-log-exp 99.1%

         \[\leadsto \left|\mathsf{fma}\left(ew, \color{blue}{\log \left(e^{\cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) \cdot \cos t}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      8. associate-*r/ 99.1%

         \[\leadsto \left|\mathsf{fma}\left(ew, \log \left(e^{\cos \tan^{-1} \color{blue}{\left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)} \cdot \cos t}\right), eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      9. cos-atan 99.1%

         \[\leadsto \left|\mathsf{fma}\left(ew, \log \left(e^{\color{blue}{\frac{1}{\sqrt{1 + \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)}}} \cdot \cos t}\right), eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

   6. Applied egg-rr 99.1%

      \[\leadsto \left|\mathsf{fma}\left(ew, \color{blue}{\log \left(e^{\frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

   7. Taylor expanded in ew around inf 78.5%

      \[\leadsto \left|\color{blue}{ew \cdot \cos t}\right| \]

   if -5.20000000000000005e-104 < ew < 6.40000000000000005e-49

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   2. Step-by-step derivation

      1. fabs-sub 99.8%

         \[\leadsto \color{blue}{\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right|} \]

      2. sub-neg 99.8%

         \[\leadsto \left|\color{blue}{\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) + \left(-\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

      3. +-commutative 99.8%

         \[\leadsto \left|\color{blue}{\left(-\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}\right| \]

      4. associate-*l* 99.8%

         \[\leadsto \left|\left(-\color{blue}{ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right) + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      5. distribute-rgt-neg-in 99.8%

         \[\leadsto \left|\color{blue}{ew \cdot \left(-\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)} + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      6. fma-define 99.8%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(ew, -\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right), \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \left(-\cos t\right), eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in ew around 0 80.7%

      \[\leadsto \left|\color{blue}{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}\right| \]
   6. Step-by-step derivation

      1. *-commutative 80.7%

         \[\leadsto \left|\color{blue}{\left(\sin t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right) \cdot eh}\right| \]

      2. associate-*l* 80.7%

         \[\leadsto \left|\color{blue}{\sin t \cdot \left(\sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right) \cdot eh\right)}\right| \]

      3. *-commutative 80.7%

         \[\leadsto \left|\sin t \cdot \color{blue}{\left(eh \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}\right| \]

      4. associate-*r/ 80.7%

         \[\leadsto \left|\sin t \cdot \left(eh \cdot \sin \tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(eh \cdot \tan t\right)}{ew}\right)}\right)\right| \]

      5. neg-mul-1 80.7%

         \[\leadsto \left|\sin t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{\color{blue}{-eh \cdot \tan t}}{ew}\right)\right)\right| \]

      6. *-commutative 80.7%

         \[\leadsto \left|\sin t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{-\color{blue}{\tan t \cdot eh}}{ew}\right)\right)\right| \]

      7. distribute-lft-neg-in 80.7%

         \[\leadsto \left|\sin t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\left(-\tan t\right) \cdot eh}}{ew}\right)\right)\right| \]

      8. associate-*r/ 80.7%

         \[\leadsto \left|\sin t \cdot \left(eh \cdot \sin \tan^{-1} \color{blue}{\left(\left(-\tan t\right) \cdot \frac{eh}{ew}\right)}\right)\right| \]

      9. *-commutative 80.7%

         \[\leadsto \left|\sin t \cdot \left(eh \cdot \sin \tan^{-1} \color{blue}{\left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right)}\right)\right| \]

   7. Simplified 80.7%

      \[\leadsto \left|\color{blue}{\sin t \cdot \left(eh \cdot \sin \tan^{-1} \left(\frac{eh}{ew} \cdot \left(-\tan t\right)\right)\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -5.2 \cdot 10^{-104} \lor \neg \left(ew \leq 6.4 \cdot 10^{-49}\right):\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sin t \cdot \left(eh \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{-eh}{ew}\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.1% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \cos t\right|\\ \mathbf{if}\;ew \leq -1.2 \cdot 10^{-133}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq -3.4 \cdot 10^{-299}:\\ \;\;\;\;\left|eh \cdot \left(t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{-eh}{ew}\right)\right)\right|\\ \mathbf{elif}\;ew \leq 3.8 \cdot 10^{-50}:\\ \;\;\;\;\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (cos t)))))
   (if (<= ew -1.2e-133)
     t_1
     (if (<= ew -3.4e-299)
       (fabs (* eh (* t (sin (atan (* (tan t) (/ (- eh) ew)))))))
       (if (<= ew 3.8e-50)
         (* (* eh (sin t)) (sin (atan (* (tan t) (/ eh ew)))))
         t_1)))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * cos(t)));
	double tmp;
	if (ew <= -1.2e-133) {
		tmp = t_1;
	} else if (ew <= -3.4e-299) {
		tmp = fabs((eh * (t * sin(atan((tan(t) * (-eh / ew)))))));
	} else if (ew <= 3.8e-50) {
		tmp = (eh * sin(t)) * sin(atan((tan(t) * (eh / ew))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((ew * cos(t)))
    if (ew <= (-1.2d-133)) then
        tmp = t_1
    else if (ew <= (-3.4d-299)) then
        tmp = abs((eh * (t * sin(atan((tan(t) * (-eh / ew)))))))
    else if (ew <= 3.8d-50) then
        tmp = (eh * sin(t)) * sin(atan((tan(t) * (eh / ew))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((ew * Math.cos(t)));
	double tmp;
	if (ew <= -1.2e-133) {
		tmp = t_1;
	} else if (ew <= -3.4e-299) {
		tmp = Math.abs((eh * (t * Math.sin(Math.atan((Math.tan(t) * (-eh / ew)))))));
	} else if (ew <= 3.8e-50) {
		tmp = (eh * Math.sin(t)) * Math.sin(Math.atan((Math.tan(t) * (eh / ew))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((ew * math.cos(t)))
	tmp = 0
	if ew <= -1.2e-133:
		tmp = t_1
	elif ew <= -3.4e-299:
		tmp = math.fabs((eh * (t * math.sin(math.atan((math.tan(t) * (-eh / ew)))))))
	elif ew <= 3.8e-50:
		tmp = (eh * math.sin(t)) * math.sin(math.atan((math.tan(t) * (eh / ew))))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * cos(t)))
	tmp = 0.0
	if (ew <= -1.2e-133)
		tmp = t_1;
	elseif (ew <= -3.4e-299)
		tmp = abs(Float64(eh * Float64(t * sin(atan(Float64(tan(t) * Float64(Float64(-eh) / ew)))))));
	elseif (ew <= 3.8e-50)
		tmp = Float64(Float64(eh * sin(t)) * sin(atan(Float64(tan(t) * Float64(eh / ew)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((ew * cos(t)));
	tmp = 0.0;
	if (ew <= -1.2e-133)
		tmp = t_1;
	elseif (ew <= -3.4e-299)
		tmp = abs((eh * (t * sin(atan((tan(t) * (-eh / ew)))))));
	elseif (ew <= 3.8e-50)
		tmp = (eh * sin(t)) * sin(atan((tan(t) * (eh / ew))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -1.2e-133], t$95$1, If[LessEqual[ew, -3.4e-299], N[Abs[N[(eh * N[(t * N[Sin[N[ArcTan[N[(N[Tan[t], $MachinePrecision] * N[((-eh) / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 3.8e-50], N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[Tan[t], $MachinePrecision] * N[(eh / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -1.2e-133 or 3.7999999999999999e-50 < ew

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   2. Step-by-step derivation

      1. fabs-sub 99.8%

         \[\leadsto \color{blue}{\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right|} \]

      2. sub-neg 99.8%

         \[\leadsto \left|\color{blue}{\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) + \left(-\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

      3. +-commutative 99.8%

         \[\leadsto \left|\color{blue}{\left(-\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}\right| \]

      4. associate-*l* 99.8%

         \[\leadsto \left|\left(-\color{blue}{ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right) + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      5. distribute-rgt-neg-in 99.8%

         \[\leadsto \left|\color{blue}{ew \cdot \left(-\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)} + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      6. fma-define 99.8%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(ew, -\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right), \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \left(-\cos t\right), eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 22.8%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \color{blue}{\left(\sqrt{-\cos t} \cdot \sqrt{-\cos t}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      2. sqrt-unprod 99.4%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \color{blue}{\sqrt{\left(-\cos t\right) \cdot \left(-\cos t\right)}}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      3. sqr-neg 99.4%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \sqrt{\color{blue}{\cos t \cdot \cos t}}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      4. sqrt-unprod 76.4%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \color{blue}{\left(\sqrt{\cos t} \cdot \sqrt{\cos t}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      5. add-sqr-sqrt 99.3%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \color{blue}{\cos t}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      6. associate-*r/ 99.3%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \color{blue}{\left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)} \cdot \cos t, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      7. add-log-exp 99.1%

         \[\leadsto \left|\mathsf{fma}\left(ew, \color{blue}{\log \left(e^{\cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) \cdot \cos t}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      8. associate-*r/ 99.1%

         \[\leadsto \left|\mathsf{fma}\left(ew, \log \left(e^{\cos \tan^{-1} \color{blue}{\left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)} \cdot \cos t}\right), eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      9. cos-atan 99.1%

         \[\leadsto \left|\mathsf{fma}\left(ew, \log \left(e^{\color{blue}{\frac{1}{\sqrt{1 + \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)}}} \cdot \cos t}\right), eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

   6. Applied egg-rr 99.1%

      \[\leadsto \left|\mathsf{fma}\left(ew, \color{blue}{\log \left(e^{\frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

   7. Taylor expanded in ew around inf 77.4%

      \[\leadsto \left|\color{blue}{ew \cdot \cos t}\right| \]

   if -1.2e-133 < ew < -3.3999999999999998e-299

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   2. Step-by-step derivation

      1. fabs-sub 99.8%

         \[\leadsto \color{blue}{\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right|} \]

      2. sub-neg 99.8%

         \[\leadsto \left|\color{blue}{\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) + \left(-\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

      3. +-commutative 99.8%

         \[\leadsto \left|\color{blue}{\left(-\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}\right| \]

      4. associate-*l* 99.8%

         \[\leadsto \left|\left(-\color{blue}{ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right) + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      5. distribute-rgt-neg-in 99.8%

         \[\leadsto \left|\color{blue}{ew \cdot \left(-\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)} + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      6. fma-define 99.8%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(ew, -\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right), \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \left(-\cos t\right), eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 70.5%

      \[\leadsto \left|\color{blue}{-1 \cdot \left(ew \cdot \cos \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right) + eh \cdot \left(t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}\right| \]

   6. Taylor expanded in ew around 0 56.2%

      \[\leadsto \left|\color{blue}{eh \cdot \left(t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}\right| \]
   7. Step-by-step derivation

      1. mul-1-neg 56.2%

         \[\leadsto \left|eh \cdot \left(t \cdot \sin \tan^{-1} \color{blue}{\left(-\frac{eh \cdot \tan t}{ew}\right)}\right)\right| \]

      2. associate-*l/ 56.2%

         \[\leadsto \left|eh \cdot \left(t \cdot \sin \tan^{-1} \left(-\color{blue}{\frac{eh}{ew} \cdot \tan t}\right)\right)\right| \]

      3. *-commutative 56.2%

         \[\leadsto \left|eh \cdot \left(t \cdot \sin \tan^{-1} \left(-\color{blue}{\tan t \cdot \frac{eh}{ew}}\right)\right)\right| \]

      4. distribute-rgt-neg-in 56.2%

         \[\leadsto \left|eh \cdot \left(t \cdot \sin \tan^{-1} \color{blue}{\left(\tan t \cdot \left(-\frac{eh}{ew}\right)\right)}\right)\right| \]

      5. distribute-frac-neg 56.2%

         \[\leadsto \left|eh \cdot \left(t \cdot \sin \tan^{-1} \left(\tan t \cdot \color{blue}{\frac{-eh}{ew}}\right)\right)\right| \]

   8. Simplified 56.2%

      \[\leadsto \left|\color{blue}{eh \cdot \left(t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{-eh}{ew}\right)\right)}\right| \]

   if -3.3999999999999998e-299 < ew < 3.7999999999999999e-50

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   2. Add Preprocessing

   4. Applied egg-rr 73.6%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(eh \cdot \sin t, \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right), \frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)}\right)}^{2}} \]

   5. Taylor expanded in eh around inf 60.4%

      \[\leadsto \color{blue}{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{ew}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*l/ 60.4%

         \[\leadsto eh \cdot \left(\sin t \cdot \sin \tan^{-1} \color{blue}{\left(\frac{eh}{ew} \cdot \tan t\right)}\right) \]

      2. *-commutative 60.4%

         \[\leadsto eh \cdot \left(\sin t \cdot \sin \tan^{-1} \color{blue}{\left(\tan t \cdot \frac{eh}{ew}\right)}\right) \]

      3. associate-*l* 60.4%

         \[\leadsto \color{blue}{\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)} \]

   7. Simplified 60.4%

      \[\leadsto \color{blue}{\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 65.2% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \cos t\right|\\ \mathbf{if}\;ew \leq -6.3 \cdot 10^{-133}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq -4.5 \cdot 10^{-300}:\\ \;\;\;\;\left|eh \cdot \left(t \cdot \sin \tan^{-1} \left(t \cdot \frac{-eh}{ew}\right)\right)\right|\\ \mathbf{elif}\;ew \leq 3.8 \cdot 10^{-50}:\\ \;\;\;\;\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (cos t)))))
   (if (<= ew -6.3e-133)
     t_1
     (if (<= ew -4.5e-300)
       (fabs (* eh (* t (sin (atan (* t (/ (- eh) ew)))))))
       (if (<= ew 3.8e-50)
         (* (* eh (sin t)) (sin (atan (* (tan t) (/ eh ew)))))
         t_1)))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * cos(t)));
	double tmp;
	if (ew <= -6.3e-133) {
		tmp = t_1;
	} else if (ew <= -4.5e-300) {
		tmp = fabs((eh * (t * sin(atan((t * (-eh / ew)))))));
	} else if (ew <= 3.8e-50) {
		tmp = (eh * sin(t)) * sin(atan((tan(t) * (eh / ew))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((ew * cos(t)))
    if (ew <= (-6.3d-133)) then
        tmp = t_1
    else if (ew <= (-4.5d-300)) then
        tmp = abs((eh * (t * sin(atan((t * (-eh / ew)))))))
    else if (ew <= 3.8d-50) then
        tmp = (eh * sin(t)) * sin(atan((tan(t) * (eh / ew))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((ew * Math.cos(t)));
	double tmp;
	if (ew <= -6.3e-133) {
		tmp = t_1;
	} else if (ew <= -4.5e-300) {
		tmp = Math.abs((eh * (t * Math.sin(Math.atan((t * (-eh / ew)))))));
	} else if (ew <= 3.8e-50) {
		tmp = (eh * Math.sin(t)) * Math.sin(Math.atan((Math.tan(t) * (eh / ew))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((ew * math.cos(t)))
	tmp = 0
	if ew <= -6.3e-133:
		tmp = t_1
	elif ew <= -4.5e-300:
		tmp = math.fabs((eh * (t * math.sin(math.atan((t * (-eh / ew)))))))
	elif ew <= 3.8e-50:
		tmp = (eh * math.sin(t)) * math.sin(math.atan((math.tan(t) * (eh / ew))))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * cos(t)))
	tmp = 0.0
	if (ew <= -6.3e-133)
		tmp = t_1;
	elseif (ew <= -4.5e-300)
		tmp = abs(Float64(eh * Float64(t * sin(atan(Float64(t * Float64(Float64(-eh) / ew)))))));
	elseif (ew <= 3.8e-50)
		tmp = Float64(Float64(eh * sin(t)) * sin(atan(Float64(tan(t) * Float64(eh / ew)))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((ew * cos(t)));
	tmp = 0.0;
	if (ew <= -6.3e-133)
		tmp = t_1;
	elseif (ew <= -4.5e-300)
		tmp = abs((eh * (t * sin(atan((t * (-eh / ew)))))));
	elseif (ew <= 3.8e-50)
		tmp = (eh * sin(t)) * sin(atan((tan(t) * (eh / ew))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -6.3e-133], t$95$1, If[LessEqual[ew, -4.5e-300], N[Abs[N[(eh * N[(t * N[Sin[N[ArcTan[N[(t * N[((-eh) / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 3.8e-50], N[(N[(eh * N[Sin[t], $MachinePrecision]), $MachinePrecision] * N[Sin[N[ArcTan[N[(N[Tan[t], $MachinePrecision] * N[(eh / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -6.3000000000000003e-133 or 3.7999999999999999e-50 < ew

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   2. Step-by-step derivation

      1. fabs-sub 99.8%

         \[\leadsto \color{blue}{\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right|} \]

      2. sub-neg 99.8%

         \[\leadsto \left|\color{blue}{\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) + \left(-\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

      3. +-commutative 99.8%

         \[\leadsto \left|\color{blue}{\left(-\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}\right| \]

      4. associate-*l* 99.8%

         \[\leadsto \left|\left(-\color{blue}{ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right) + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      5. distribute-rgt-neg-in 99.8%

         \[\leadsto \left|\color{blue}{ew \cdot \left(-\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)} + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      6. fma-define 99.8%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(ew, -\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right), \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \left(-\cos t\right), eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 22.8%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \color{blue}{\left(\sqrt{-\cos t} \cdot \sqrt{-\cos t}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      2. sqrt-unprod 99.4%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \color{blue}{\sqrt{\left(-\cos t\right) \cdot \left(-\cos t\right)}}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      3. sqr-neg 99.4%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \sqrt{\color{blue}{\cos t \cdot \cos t}}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      4. sqrt-unprod 76.4%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \color{blue}{\left(\sqrt{\cos t} \cdot \sqrt{\cos t}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      5. add-sqr-sqrt 99.3%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \color{blue}{\cos t}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      6. associate-*r/ 99.3%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \color{blue}{\left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)} \cdot \cos t, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      7. add-log-exp 99.1%

         \[\leadsto \left|\mathsf{fma}\left(ew, \color{blue}{\log \left(e^{\cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) \cdot \cos t}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      8. associate-*r/ 99.1%

         \[\leadsto \left|\mathsf{fma}\left(ew, \log \left(e^{\cos \tan^{-1} \color{blue}{\left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)} \cdot \cos t}\right), eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      9. cos-atan 99.1%

         \[\leadsto \left|\mathsf{fma}\left(ew, \log \left(e^{\color{blue}{\frac{1}{\sqrt{1 + \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)}}} \cdot \cos t}\right), eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

   6. Applied egg-rr 99.1%

      \[\leadsto \left|\mathsf{fma}\left(ew, \color{blue}{\log \left(e^{\frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

   7. Taylor expanded in ew around inf 77.4%

      \[\leadsto \left|\color{blue}{ew \cdot \cos t}\right| \]

   if -6.3000000000000003e-133 < ew < -4.5e-300

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   2. Step-by-step derivation

      1. fabs-sub 99.8%

         \[\leadsto \color{blue}{\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right|} \]

      2. sub-neg 99.8%

         \[\leadsto \left|\color{blue}{\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) + \left(-\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

      3. +-commutative 99.8%

         \[\leadsto \left|\color{blue}{\left(-\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}\right| \]

      4. associate-*l* 99.8%

         \[\leadsto \left|\left(-\color{blue}{ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right) + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      5. distribute-rgt-neg-in 99.8%

         \[\leadsto \left|\color{blue}{ew \cdot \left(-\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)} + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      6. fma-define 99.8%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(ew, -\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right), \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \left(-\cos t\right), eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 70.5%

      \[\leadsto \left|\color{blue}{-1 \cdot \left(ew \cdot \cos \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right) + eh \cdot \left(t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}\right| \]

   6. Taylor expanded in ew around 0 56.2%

      \[\leadsto \left|\color{blue}{eh \cdot \left(t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}\right| \]
   7. Step-by-step derivation

      1. mul-1-neg 56.2%

         \[\leadsto \left|eh \cdot \left(t \cdot \sin \tan^{-1} \color{blue}{\left(-\frac{eh \cdot \tan t}{ew}\right)}\right)\right| \]

      2. associate-*l/ 56.2%

         \[\leadsto \left|eh \cdot \left(t \cdot \sin \tan^{-1} \left(-\color{blue}{\frac{eh}{ew} \cdot \tan t}\right)\right)\right| \]

      3. *-commutative 56.2%

         \[\leadsto \left|eh \cdot \left(t \cdot \sin \tan^{-1} \left(-\color{blue}{\tan t \cdot \frac{eh}{ew}}\right)\right)\right| \]

      4. distribute-rgt-neg-in 56.2%

         \[\leadsto \left|eh \cdot \left(t \cdot \sin \tan^{-1} \color{blue}{\left(\tan t \cdot \left(-\frac{eh}{ew}\right)\right)}\right)\right| \]

      5. distribute-frac-neg 56.2%

         \[\leadsto \left|eh \cdot \left(t \cdot \sin \tan^{-1} \left(\tan t \cdot \color{blue}{\frac{-eh}{ew}}\right)\right)\right| \]

   8. Simplified 56.2%

      \[\leadsto \left|\color{blue}{eh \cdot \left(t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{-eh}{ew}\right)\right)}\right| \]

   9. Taylor expanded in t around 0 56.2%

      \[\leadsto \left|eh \cdot \left(t \cdot \sin \tan^{-1} \color{blue}{\left(-1 \cdot \frac{eh \cdot t}{ew}\right)}\right)\right| \]
   10. Step-by-step derivation

       1. associate-*r/ 56.2%

          \[\leadsto \left|eh \cdot \left(t \cdot \sin \tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(eh \cdot t\right)}{ew}\right)}\right)\right| \]

       2. associate-*r* 56.2%

          \[\leadsto \left|eh \cdot \left(t \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\left(-1 \cdot eh\right) \cdot t}}{ew}\right)\right)\right| \]

       3. neg-mul-1 56.2%

          \[\leadsto \left|eh \cdot \left(t \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\left(-eh\right)} \cdot t}{ew}\right)\right)\right| \]

       4. associate-*l/ 56.2%

          \[\leadsto \left|eh \cdot \left(t \cdot \sin \tan^{-1} \color{blue}{\left(\frac{-eh}{ew} \cdot t\right)}\right)\right| \]

       5. *-commutative 56.2%

          \[\leadsto \left|eh \cdot \left(t \cdot \sin \tan^{-1} \color{blue}{\left(t \cdot \frac{-eh}{ew}\right)}\right)\right| \]

   11. Simplified 56.2%

       \[\leadsto \left|eh \cdot \left(t \cdot \sin \tan^{-1} \color{blue}{\left(t \cdot \frac{-eh}{ew}\right)}\right)\right| \]

   if -4.5e-300 < ew < 3.7999999999999999e-50

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   2. Add Preprocessing

   4. Applied egg-rr 73.6%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(eh \cdot \sin t, \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right), \frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)}\right)}^{2}} \]

   5. Taylor expanded in eh around inf 60.4%

      \[\leadsto \color{blue}{eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{ew}\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*l/ 60.4%

         \[\leadsto eh \cdot \left(\sin t \cdot \sin \tan^{-1} \color{blue}{\left(\frac{eh}{ew} \cdot \tan t\right)}\right) \]

      2. *-commutative 60.4%

         \[\leadsto eh \cdot \left(\sin t \cdot \sin \tan^{-1} \color{blue}{\left(\tan t \cdot \frac{eh}{ew}\right)}\right) \]

      3. associate-*l* 60.4%

         \[\leadsto \color{blue}{\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)} \]

   7. Simplified 60.4%

      \[\leadsto \color{blue}{\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 65.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left|ew \cdot \cos t\right|\\ \mathbf{if}\;ew \leq -8.8 \cdot 10^{-134}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;ew \leq -4 \cdot 10^{-310}:\\ \;\;\;\;\left|eh \cdot \left(t \cdot \sin \tan^{-1} \left(t \cdot \frac{-eh}{ew}\right)\right)\right|\\ \mathbf{elif}\;ew \leq 8.2 \cdot 10^{-75}:\\ \;\;\;\;ew + eh \cdot \left(t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{ew}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (let* ((t_1 (fabs (* ew (cos t)))))
   (if (<= ew -8.8e-134)
     t_1
     (if (<= ew -4e-310)
       (fabs (* eh (* t (sin (atan (* t (/ (- eh) ew)))))))
       (if (<= ew 8.2e-75)
         (+ ew (* eh (* t (sin (atan (/ (* eh (tan t)) ew))))))
         t_1)))))

C

double code(double eh, double ew, double t) {
	double t_1 = fabs((ew * cos(t)));
	double tmp;
	if (ew <= -8.8e-134) {
		tmp = t_1;
	} else if (ew <= -4e-310) {
		tmp = fabs((eh * (t * sin(atan((t * (-eh / ew)))))));
	} else if (ew <= 8.2e-75) {
		tmp = ew + (eh * (t * sin(atan(((eh * tan(t)) / ew)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = abs((ew * cos(t)))
    if (ew <= (-8.8d-134)) then
        tmp = t_1
    else if (ew <= (-4d-310)) then
        tmp = abs((eh * (t * sin(atan((t * (-eh / ew)))))))
    else if (ew <= 8.2d-75) then
        tmp = ew + (eh * (t * sin(atan(((eh * tan(t)) / ew)))))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double t_1 = Math.abs((ew * Math.cos(t)));
	double tmp;
	if (ew <= -8.8e-134) {
		tmp = t_1;
	} else if (ew <= -4e-310) {
		tmp = Math.abs((eh * (t * Math.sin(Math.atan((t * (-eh / ew)))))));
	} else if (ew <= 8.2e-75) {
		tmp = ew + (eh * (t * Math.sin(Math.atan(((eh * Math.tan(t)) / ew)))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(eh, ew, t):
	t_1 = math.fabs((ew * math.cos(t)))
	tmp = 0
	if ew <= -8.8e-134:
		tmp = t_1
	elif ew <= -4e-310:
		tmp = math.fabs((eh * (t * math.sin(math.atan((t * (-eh / ew)))))))
	elif ew <= 8.2e-75:
		tmp = ew + (eh * (t * math.sin(math.atan(((eh * math.tan(t)) / ew)))))
	else:
		tmp = t_1
	return tmp

Julia

function code(eh, ew, t)
	t_1 = abs(Float64(ew * cos(t)))
	tmp = 0.0
	if (ew <= -8.8e-134)
		tmp = t_1;
	elseif (ew <= -4e-310)
		tmp = abs(Float64(eh * Float64(t * sin(atan(Float64(t * Float64(Float64(-eh) / ew)))))));
	elseif (ew <= 8.2e-75)
		tmp = Float64(ew + Float64(eh * Float64(t * sin(atan(Float64(Float64(eh * tan(t)) / ew))))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	t_1 = abs((ew * cos(t)));
	tmp = 0.0;
	if (ew <= -8.8e-134)
		tmp = t_1;
	elseif (ew <= -4e-310)
		tmp = abs((eh * (t * sin(atan((t * (-eh / ew)))))));
	elseif (ew <= 8.2e-75)
		tmp = ew + (eh * (t * sin(atan(((eh * tan(t)) / ew)))));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := Block[{t$95$1 = N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[ew, -8.8e-134], t$95$1, If[LessEqual[ew, -4e-310], N[Abs[N[(eh * N[(t * N[Sin[N[ArcTan[N[(t * N[((-eh) / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[ew, 8.2e-75], N[(ew + N[(eh * N[(t * N[Sin[N[ArcTan[N[(N[(eh * N[Tan[t], $MachinePrecision]), $MachinePrecision] / ew), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if ew < -8.7999999999999999e-134 or 8.20000000000000005e-75 < ew

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   2. Step-by-step derivation

      1. fabs-sub 99.8%

         \[\leadsto \color{blue}{\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right|} \]

      2. sub-neg 99.8%

         \[\leadsto \left|\color{blue}{\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) + \left(-\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

      3. +-commutative 99.8%

         \[\leadsto \left|\color{blue}{\left(-\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}\right| \]

      4. associate-*l* 99.8%

         \[\leadsto \left|\left(-\color{blue}{ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right) + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      5. distribute-rgt-neg-in 99.8%

         \[\leadsto \left|\color{blue}{ew \cdot \left(-\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)} + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      6. fma-define 99.8%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(ew, -\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right), \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \left(-\cos t\right), eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 23.0%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \color{blue}{\left(\sqrt{-\cos t} \cdot \sqrt{-\cos t}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      2. sqrt-unprod 99.4%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \color{blue}{\sqrt{\left(-\cos t\right) \cdot \left(-\cos t\right)}}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      3. sqr-neg 99.4%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \sqrt{\color{blue}{\cos t \cdot \cos t}}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      4. sqrt-unprod 76.2%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \color{blue}{\left(\sqrt{\cos t} \cdot \sqrt{\cos t}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      5. add-sqr-sqrt 99.3%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \color{blue}{\cos t}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      6. associate-*r/ 99.3%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \color{blue}{\left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)} \cdot \cos t, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      7. add-log-exp 99.1%

         \[\leadsto \left|\mathsf{fma}\left(ew, \color{blue}{\log \left(e^{\cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) \cdot \cos t}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      8. associate-*r/ 99.1%

         \[\leadsto \left|\mathsf{fma}\left(ew, \log \left(e^{\cos \tan^{-1} \color{blue}{\left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)} \cdot \cos t}\right), eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      9. cos-atan 99.1%

         \[\leadsto \left|\mathsf{fma}\left(ew, \log \left(e^{\color{blue}{\frac{1}{\sqrt{1 + \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)}}} \cdot \cos t}\right), eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

   6. Applied egg-rr 99.1%

      \[\leadsto \left|\mathsf{fma}\left(ew, \color{blue}{\log \left(e^{\frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

   7. Taylor expanded in ew around inf 76.0%

      \[\leadsto \left|\color{blue}{ew \cdot \cos t}\right| \]

   if -8.7999999999999999e-134 < ew < -3.999999999999988e-310

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   2. Step-by-step derivation

      1. fabs-sub 99.8%

         \[\leadsto \color{blue}{\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right|} \]

      2. sub-neg 99.8%

         \[\leadsto \left|\color{blue}{\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) + \left(-\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

      3. +-commutative 99.8%

         \[\leadsto \left|\color{blue}{\left(-\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}\right| \]

      4. associate-*l* 99.8%

         \[\leadsto \left|\left(-\color{blue}{ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right) + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      5. distribute-rgt-neg-in 99.8%

         \[\leadsto \left|\color{blue}{ew \cdot \left(-\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)} + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      6. fma-define 99.8%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(ew, -\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right), \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \left(-\cos t\right), eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 69.7%

      \[\leadsto \left|\color{blue}{-1 \cdot \left(ew \cdot \cos \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right) + eh \cdot \left(t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}\right| \]

   6. Taylor expanded in ew around 0 53.9%

      \[\leadsto \left|\color{blue}{eh \cdot \left(t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}\right| \]
   7. Step-by-step derivation

      1. mul-1-neg 53.9%

         \[\leadsto \left|eh \cdot \left(t \cdot \sin \tan^{-1} \color{blue}{\left(-\frac{eh \cdot \tan t}{ew}\right)}\right)\right| \]

      2. associate-*l/ 53.9%

         \[\leadsto \left|eh \cdot \left(t \cdot \sin \tan^{-1} \left(-\color{blue}{\frac{eh}{ew} \cdot \tan t}\right)\right)\right| \]

      3. *-commutative 53.9%

         \[\leadsto \left|eh \cdot \left(t \cdot \sin \tan^{-1} \left(-\color{blue}{\tan t \cdot \frac{eh}{ew}}\right)\right)\right| \]

      4. distribute-rgt-neg-in 53.9%

         \[\leadsto \left|eh \cdot \left(t \cdot \sin \tan^{-1} \color{blue}{\left(\tan t \cdot \left(-\frac{eh}{ew}\right)\right)}\right)\right| \]

      5. distribute-frac-neg 53.9%

         \[\leadsto \left|eh \cdot \left(t \cdot \sin \tan^{-1} \left(\tan t \cdot \color{blue}{\frac{-eh}{ew}}\right)\right)\right| \]

   8. Simplified 53.9%

      \[\leadsto \left|\color{blue}{eh \cdot \left(t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{-eh}{ew}\right)\right)}\right| \]

   9. Taylor expanded in t around 0 53.9%

      \[\leadsto \left|eh \cdot \left(t \cdot \sin \tan^{-1} \color{blue}{\left(-1 \cdot \frac{eh \cdot t}{ew}\right)}\right)\right| \]
   10. Step-by-step derivation

       1. associate-*r/ 53.9%

          \[\leadsto \left|eh \cdot \left(t \cdot \sin \tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(eh \cdot t\right)}{ew}\right)}\right)\right| \]

       2. associate-*r* 53.9%

          \[\leadsto \left|eh \cdot \left(t \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\left(-1 \cdot eh\right) \cdot t}}{ew}\right)\right)\right| \]

       3. neg-mul-1 53.9%

          \[\leadsto \left|eh \cdot \left(t \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\left(-eh\right)} \cdot t}{ew}\right)\right)\right| \]

       4. associate-*l/ 53.9%

          \[\leadsto \left|eh \cdot \left(t \cdot \sin \tan^{-1} \color{blue}{\left(\frac{-eh}{ew} \cdot t\right)}\right)\right| \]

       5. *-commutative 53.9%

          \[\leadsto \left|eh \cdot \left(t \cdot \sin \tan^{-1} \color{blue}{\left(t \cdot \frac{-eh}{ew}\right)}\right)\right| \]

   11. Simplified 53.9%

       \[\leadsto \left|eh \cdot \left(t \cdot \sin \tan^{-1} \color{blue}{\left(t \cdot \frac{-eh}{ew}\right)}\right)\right| \]

   if -3.999999999999988e-310 < ew < 8.20000000000000005e-75

   1. Initial program 99.9%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   2. Add Preprocessing

   4. Applied egg-rr 75.0%

      \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(eh \cdot \sin t, \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right), \frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)}\right)}^{2}} \]

   5. Taylor expanded in t around 0 49.5%

      \[\leadsto \color{blue}{ew + eh \cdot \left(t \cdot \sin \tan^{-1} \left(\frac{eh \cdot \tan t}{ew}\right)\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 63.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;ew \leq -2.7 \cdot 10^{-132} \lor \neg \left(ew \leq 7 \cdot 10^{-184}\right):\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \left(t \cdot \sin \tan^{-1} \left(t \cdot \frac{-eh}{ew}\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (eh ew t)
 :precision binary64
 (if (or (<= ew -2.7e-132) (not (<= ew 7e-184)))
   (fabs (* ew (cos t)))
   (fabs (* eh (* t (sin (atan (* t (/ (- eh) ew)))))))))

C

double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -2.7e-132) || !(ew <= 7e-184)) {
		tmp = fabs((ew * cos(t)));
	} else {
		tmp = fabs((eh * (t * sin(atan((t * (-eh / ew)))))));
	}
	return tmp;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((ew <= (-2.7d-132)) .or. (.not. (ew <= 7d-184))) then
        tmp = abs((ew * cos(t)))
    else
        tmp = abs((eh * (t * sin(atan((t * (-eh / ew)))))))
    end if
    code = tmp
end function

Java

public static double code(double eh, double ew, double t) {
	double tmp;
	if ((ew <= -2.7e-132) || !(ew <= 7e-184)) {
		tmp = Math.abs((ew * Math.cos(t)));
	} else {
		tmp = Math.abs((eh * (t * Math.sin(Math.atan((t * (-eh / ew)))))));
	}
	return tmp;
}

Python

def code(eh, ew, t):
	tmp = 0
	if (ew <= -2.7e-132) or not (ew <= 7e-184):
		tmp = math.fabs((ew * math.cos(t)))
	else:
		tmp = math.fabs((eh * (t * math.sin(math.atan((t * (-eh / ew)))))))
	return tmp

Julia

function code(eh, ew, t)
	tmp = 0.0
	if ((ew <= -2.7e-132) || !(ew <= 7e-184))
		tmp = abs(Float64(ew * cos(t)));
	else
		tmp = abs(Float64(eh * Float64(t * sin(atan(Float64(t * Float64(Float64(-eh) / ew)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(eh, ew, t)
	tmp = 0.0;
	if ((ew <= -2.7e-132) || ~((ew <= 7e-184)))
		tmp = abs((ew * cos(t)));
	else
		tmp = abs((eh * (t * sin(atan((t * (-eh / ew)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[eh_, ew_, t_] := If[Or[LessEqual[ew, -2.7e-132], N[Not[LessEqual[ew, 7e-184]], $MachinePrecision]], N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(eh * N[(t * N[Sin[N[ArcTan[N[(t * N[((-eh) / ew), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if ew < -2.6999999999999999e-132 or 6.99999999999999962e-184 < ew

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   2. Step-by-step derivation

      1. fabs-sub 99.8%

         \[\leadsto \color{blue}{\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right|} \]

      2. sub-neg 99.8%

         \[\leadsto \left|\color{blue}{\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) + \left(-\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

      3. +-commutative 99.8%

         \[\leadsto \left|\color{blue}{\left(-\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}\right| \]

      4. associate-*l* 99.8%

         \[\leadsto \left|\left(-\color{blue}{ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right) + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      5. distribute-rgt-neg-in 99.8%

         \[\leadsto \left|\color{blue}{ew \cdot \left(-\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)} + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      6. fma-define 99.8%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(ew, -\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right), \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \left(-\cos t\right), eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right|} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-sqr-sqrt 23.6%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \color{blue}{\left(\sqrt{-\cos t} \cdot \sqrt{-\cos t}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      2. sqrt-unprod 99.4%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \color{blue}{\sqrt{\left(-\cos t\right) \cdot \left(-\cos t\right)}}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      3. sqr-neg 99.4%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \sqrt{\color{blue}{\cos t \cdot \cos t}}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      4. sqrt-unprod 75.7%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \color{blue}{\left(\sqrt{\cos t} \cdot \sqrt{\cos t}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      5. add-sqr-sqrt 99.4%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \color{blue}{\cos t}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      6. associate-*r/ 99.4%

         \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \color{blue}{\left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)} \cdot \cos t, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      7. add-log-exp 99.2%

         \[\leadsto \left|\mathsf{fma}\left(ew, \color{blue}{\log \left(e^{\cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) \cdot \cos t}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      8. associate-*r/ 99.2%

         \[\leadsto \left|\mathsf{fma}\left(ew, \log \left(e^{\cos \tan^{-1} \color{blue}{\left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)} \cdot \cos t}\right), eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

      9. cos-atan 99.2%

         \[\leadsto \left|\mathsf{fma}\left(ew, \log \left(e^{\color{blue}{\frac{1}{\sqrt{1 + \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)}}} \cdot \cos t}\right), eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

   6. Applied egg-rr 99.2%

      \[\leadsto \left|\mathsf{fma}\left(ew, \color{blue}{\log \left(e^{\frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

   7. Taylor expanded in ew around inf 72.5%

      \[\leadsto \left|\color{blue}{ew \cdot \cos t}\right| \]

   if -2.6999999999999999e-132 < ew < 6.99999999999999962e-184

   1. Initial program 99.8%

      \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
   2. Step-by-step derivation

      1. fabs-sub 99.8%

         \[\leadsto \color{blue}{\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right|} \]

      2. sub-neg 99.8%

         \[\leadsto \left|\color{blue}{\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) + \left(-\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

      3. +-commutative 99.8%

         \[\leadsto \left|\color{blue}{\left(-\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}\right| \]

      4. associate-*l* 99.8%

         \[\leadsto \left|\left(-\color{blue}{ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right) + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      5. distribute-rgt-neg-in 99.8%

         \[\leadsto \left|\color{blue}{ew \cdot \left(-\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)} + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

      6. fma-define 99.8%

         \[\leadsto \left|\color{blue}{\mathsf{fma}\left(ew, -\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right), \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \left(-\cos t\right), eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right|} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 61.4%

      \[\leadsto \left|\color{blue}{-1 \cdot \left(ew \cdot \cos \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right) + eh \cdot \left(t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}\right| \]

   6. Taylor expanded in ew around 0 48.4%

      \[\leadsto \left|\color{blue}{eh \cdot \left(t \cdot \sin \tan^{-1} \left(-1 \cdot \frac{eh \cdot \tan t}{ew}\right)\right)}\right| \]
   7. Step-by-step derivation

      1. mul-1-neg 48.4%

         \[\leadsto \left|eh \cdot \left(t \cdot \sin \tan^{-1} \color{blue}{\left(-\frac{eh \cdot \tan t}{ew}\right)}\right)\right| \]

      2. associate-*l/ 48.4%

         \[\leadsto \left|eh \cdot \left(t \cdot \sin \tan^{-1} \left(-\color{blue}{\frac{eh}{ew} \cdot \tan t}\right)\right)\right| \]

      3. *-commutative 48.4%

         \[\leadsto \left|eh \cdot \left(t \cdot \sin \tan^{-1} \left(-\color{blue}{\tan t \cdot \frac{eh}{ew}}\right)\right)\right| \]

      4. distribute-rgt-neg-in 48.4%

         \[\leadsto \left|eh \cdot \left(t \cdot \sin \tan^{-1} \color{blue}{\left(\tan t \cdot \left(-\frac{eh}{ew}\right)\right)}\right)\right| \]

      5. distribute-frac-neg 48.4%

         \[\leadsto \left|eh \cdot \left(t \cdot \sin \tan^{-1} \left(\tan t \cdot \color{blue}{\frac{-eh}{ew}}\right)\right)\right| \]

   8. Simplified 48.4%

      \[\leadsto \left|\color{blue}{eh \cdot \left(t \cdot \sin \tan^{-1} \left(\tan t \cdot \frac{-eh}{ew}\right)\right)}\right| \]

   9. Taylor expanded in t around 0 48.4%

      \[\leadsto \left|eh \cdot \left(t \cdot \sin \tan^{-1} \color{blue}{\left(-1 \cdot \frac{eh \cdot t}{ew}\right)}\right)\right| \]
   10. Step-by-step derivation

       1. associate-*r/ 48.4%

          \[\leadsto \left|eh \cdot \left(t \cdot \sin \tan^{-1} \color{blue}{\left(\frac{-1 \cdot \left(eh \cdot t\right)}{ew}\right)}\right)\right| \]

       2. associate-*r* 48.4%

          \[\leadsto \left|eh \cdot \left(t \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\left(-1 \cdot eh\right) \cdot t}}{ew}\right)\right)\right| \]

       3. neg-mul-1 48.4%

          \[\leadsto \left|eh \cdot \left(t \cdot \sin \tan^{-1} \left(\frac{\color{blue}{\left(-eh\right)} \cdot t}{ew}\right)\right)\right| \]

       4. associate-*l/ 48.4%

          \[\leadsto \left|eh \cdot \left(t \cdot \sin \tan^{-1} \color{blue}{\left(\frac{-eh}{ew} \cdot t\right)}\right)\right| \]

       5. *-commutative 48.4%

          \[\leadsto \left|eh \cdot \left(t \cdot \sin \tan^{-1} \color{blue}{\left(t \cdot \frac{-eh}{ew}\right)}\right)\right| \]

   11. Simplified 48.4%

       \[\leadsto \left|eh \cdot \left(t \cdot \sin \tan^{-1} \color{blue}{\left(t \cdot \frac{-eh}{ew}\right)}\right)\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;ew \leq -2.7 \cdot 10^{-132} \lor \neg \left(ew \leq 7 \cdot 10^{-184}\right):\\ \;\;\;\;\left|ew \cdot \cos t\right|\\ \mathbf{else}:\\ \;\;\;\;\left|eh \cdot \left(t \cdot \sin \tan^{-1} \left(t \cdot \frac{-eh}{ew}\right)\right)\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 61.2% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \left|ew \cdot \cos t\right| \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (fabs (* ew (cos t))))

C

double code(double eh, double ew, double t) {
	return fabs((ew * cos(t)));
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = abs((ew * cos(t)))
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs((ew * Math.cos(t)));
}

Python

def code(eh, ew, t):
	return math.fabs((ew * math.cos(t)))

Julia

function code(eh, ew, t)
	return abs(Float64(ew * cos(t)))
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs((ew * cos(t)));
end

Wolfram

code[eh_, ew_, t_] := N[Abs[N[(ew * N[Cos[t], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
2. Step-by-step derivation

   1. fabs-sub 99.8%

      \[\leadsto \color{blue}{\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right|} \]

   2. sub-neg 99.8%

      \[\leadsto \left|\color{blue}{\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) + \left(-\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

   3. +-commutative 99.8%

      \[\leadsto \left|\color{blue}{\left(-\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}\right| \]

   4. associate-*l* 99.8%

      \[\leadsto \left|\left(-\color{blue}{ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right) + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   5. distribute-rgt-neg-in 99.8%

      \[\leadsto \left|\color{blue}{ew \cdot \left(-\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)} + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   6. fma-define 99.8%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(ew, -\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right), \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \left(-\cos t\right), eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right|} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 21.8%

      \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \color{blue}{\left(\sqrt{-\cos t} \cdot \sqrt{-\cos t}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

   2. sqrt-unprod 99.3%

      \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \color{blue}{\sqrt{\left(-\cos t\right) \cdot \left(-\cos t\right)}}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

   3. sqr-neg 99.3%

      \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \sqrt{\color{blue}{\cos t \cdot \cos t}}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

   4. sqrt-unprod 77.4%

      \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \color{blue}{\left(\sqrt{\cos t} \cdot \sqrt{\cos t}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

   5. add-sqr-sqrt 99.2%

      \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \color{blue}{\cos t}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

   6. associate-*r/ 99.2%

      \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \color{blue}{\left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)} \cdot \cos t, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

   7. add-log-exp 99.1%

      \[\leadsto \left|\mathsf{fma}\left(ew, \color{blue}{\log \left(e^{\cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) \cdot \cos t}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

   8. associate-*r/ 99.1%

      \[\leadsto \left|\mathsf{fma}\left(ew, \log \left(e^{\cos \tan^{-1} \color{blue}{\left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)} \cdot \cos t}\right), eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

   9. cos-atan 99.1%

      \[\leadsto \left|\mathsf{fma}\left(ew, \log \left(e^{\color{blue}{\frac{1}{\sqrt{1 + \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)}}} \cdot \cos t}\right), eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

6. Applied egg-rr 99.1%

   \[\leadsto \left|\mathsf{fma}\left(ew, \color{blue}{\log \left(e^{\frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

7. Taylor expanded in ew around inf 57.1%

   \[\leadsto \left|\color{blue}{ew \cdot \cos t}\right| \]
8. Add Preprocessing

Alternative 13: 41.2% accurate, 9.1× speedup

Math

\[\begin{array}{l} \\ \left|ew\right| \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 (fabs ew))

C

double code(double eh, double ew, double t) {
	return fabs(ew);
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = abs(ew)
end function

Java

public static double code(double eh, double ew, double t) {
	return Math.abs(ew);
}

Python

def code(eh, ew, t):
	return math.fabs(ew)

Julia

function code(eh, ew, t)
	return abs(ew)
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = abs(ew);
end

Wolfram

code[eh_, ew_, t_] := N[Abs[ew], $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
2. Step-by-step derivation

   1. fabs-sub 99.8%

      \[\leadsto \color{blue}{\left|\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right|} \]

   2. sub-neg 99.8%

      \[\leadsto \left|\color{blue}{\left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) + \left(-\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

   3. +-commutative 99.8%

      \[\leadsto \left|\color{blue}{\left(-\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right) + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)}\right| \]

   4. associate-*l* 99.8%

      \[\leadsto \left|\left(-\color{blue}{ew \cdot \left(\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right) + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   5. distribute-rgt-neg-in 99.8%

      \[\leadsto \left|\color{blue}{ew \cdot \left(-\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)} + \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]

   6. fma-define 99.8%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(ew, -\cos t \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right), \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right)}\right| \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \left(-\cos t\right), eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right|} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-sqr-sqrt 21.8%

      \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \color{blue}{\left(\sqrt{-\cos t} \cdot \sqrt{-\cos t}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

   2. sqrt-unprod 99.3%

      \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \color{blue}{\sqrt{\left(-\cos t\right) \cdot \left(-\cos t\right)}}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

   3. sqr-neg 99.3%

      \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \sqrt{\color{blue}{\cos t \cdot \cos t}}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

   4. sqrt-unprod 77.4%

      \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \color{blue}{\left(\sqrt{\cos t} \cdot \sqrt{\cos t}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

   5. add-sqr-sqrt 99.2%

      \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \color{blue}{\cos t}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

   6. associate-*r/ 99.2%

      \[\leadsto \left|\mathsf{fma}\left(ew, \cos \tan^{-1} \color{blue}{\left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)} \cdot \cos t, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

   7. add-log-exp 99.1%

      \[\leadsto \left|\mathsf{fma}\left(ew, \color{blue}{\log \left(e^{\cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) \cdot \cos t}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

   8. associate-*r/ 99.1%

      \[\leadsto \left|\mathsf{fma}\left(ew, \log \left(e^{\cos \tan^{-1} \color{blue}{\left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)} \cdot \cos t}\right), eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

   9. cos-atan 99.1%

      \[\leadsto \left|\mathsf{fma}\left(ew, \log \left(e^{\color{blue}{\frac{1}{\sqrt{1 + \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right) \cdot \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)}}} \cdot \cos t}\right), eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

6. Applied egg-rr 99.1%

   \[\leadsto \left|\mathsf{fma}\left(ew, \color{blue}{\log \left(e^{\frac{\cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}}\right)}, eh \cdot \left(\sin t \cdot \sin \tan^{-1} \left(\left(-eh\right) \cdot \frac{\tan t}{ew}\right)\right)\right)\right| \]

7. Taylor expanded in t around 0 40.7%

   \[\leadsto \left|\color{blue}{ew}\right| \]
8. Add Preprocessing

Alternative 14: 21.7% accurate, 921.0× speedup

Math

\[\begin{array}{l} \\ ew \end{array} \]

FPCore

(FPCore (eh ew t) :precision binary64 ew)

C

double code(double eh, double ew, double t) {
	return ew;
}

Fortran

real(8) function code(eh, ew, t)
    real(8), intent (in) :: eh
    real(8), intent (in) :: ew
    real(8), intent (in) :: t
    code = ew
end function

Java

public static double code(double eh, double ew, double t) {
	return ew;
}

Python

def code(eh, ew, t):
	return ew

Julia

function code(eh, ew, t)
	return ew
end

MATLAB

function tmp = code(eh, ew, t)
	tmp = ew;
end

Wolfram

code[eh_, ew_, t_] := ew

Derivation

1. Initial program 99.8%

   \[\left|\left(ew \cdot \cos t\right) \cdot \cos \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right) - \left(eh \cdot \sin t\right) \cdot \sin \tan^{-1} \left(\frac{\left(-eh\right) \cdot \tan t}{ew}\right)\right| \]
2. Add Preprocessing

4. Applied egg-rr 48.0%

   \[\leadsto \color{blue}{{\left(\sqrt{\mathsf{fma}\left(eh \cdot \sin t, \sin \tan^{-1} \left(\tan t \cdot \frac{eh}{ew}\right), \frac{ew \cdot \cos t}{\mathsf{hypot}\left(1, \tan t \cdot \frac{eh}{ew}\right)}\right)}\right)}^{2}} \]

5. Taylor expanded in t around 0 18.6%

   \[\leadsto {\color{blue}{\left(\sqrt{ew}\right)}}^{2} \]
6. Step-by-step derivation

   1. *-un-lft-identity 18.6%

      \[\leadsto \color{blue}{1 \cdot {\left(\sqrt{ew}\right)}^{2}} \]

   2. unpow2 18.6%

      \[\leadsto 1 \cdot \color{blue}{\left(\sqrt{ew} \cdot \sqrt{ew}\right)} \]

   3. add-sqr-sqrt 19.6%

      \[\leadsto 1 \cdot \color{blue}{ew} \]

7. Applied egg-rr 19.6%

   \[\leadsto \color{blue}{1 \cdot ew} \]

8. Final simplification 19.6%

   \[\leadsto ew \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024185 
(FPCore (eh ew t)
  :name "Example 2 from Robby"
  :precision binary64
  (fabs (- (* (* ew (cos t)) (cos (atan (/ (* (- eh) (tan t)) ew)))) (* (* eh (sin t)) (sin (atan (/ (* (- eh) (tan t)) ew)))))))